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Triangles Diagonal Factorial

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Triangles with a column or diagonal the factorial numbers

A008275 0 FC Triangle read by rows of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.

A008276 0 LD Triangle of Stirling numbers of first kind, s(n,n-k+1), n >= 1, 1<=k<=n. Also triangle T(n,k) giving coefficients in expansion of n!*C(x,n)/x in powers of x.

A008279 0 LD Triangle T(n,k) = n!/(n-k)! (0 <= k <= n) read by rows, giving number of permutations of n things k at a time.

A008279 0 SD Triangle T(n,k) = n!/(n-k)! (0 <= k <= n) read by rows, giving number of permutations of n things k at a time.

A008285 0 LD Triangle T(n,k) = number of labeled order relations on n nodes in which longest chain has k nodes (n >= 1, 1 <= k <= n).

A008296 0 FC Triangle of Lehmer-Comtet numbers of first kind.

A008297 0 FC Triangle of Lah numbers.

A008300 0 SC Triangle read by rows: T(n,k) (n >= 0, 0<=k<=n) gives number of {0,1} n X n matrices with all row and column sums equal to k.

A008300 0 SD Triangle read by rows: T(n,k) (n >= 0, 0<=k<=n) gives number of {0,1} n X n matrices with all row and column sums equal to k.

A008305 0 LD Triangle read by rows: a(n,k) = number of permutations of [n] allowing i->i+j (mod n), j=0..k-1.

A008517 0 LD Second-order Eulerian triangle T(n,k), 1<=k<=n.

A009963 0 SC Triangle of numbers n!(n-1)!...(n-k+1)!/1!2!...k!.

A009963 0 SD Triangle of numbers n!(n-1)!...(n-k+1)!/1!2!...k!.

A019538 0 LD Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).

A019575 0 FC Place n distinguishable balls in n boxes (in n^n ways); let T(n,k) = number of ways that the maximum in any box is k, for 1<=k<=n; sequence gives triangle of numbers T(n,k).

A019576 0 FC Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives triangle of numbers f(n,k)/n.

A021009 0 FC Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).

A021010 0 LD Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).

A021012 0 FC Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).

A021012 0 LD Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).

A028246 0 LD Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*C(k,i)*i^n; n >= 1, 1<=k<=n, read by rows.

A028421 0 FC Array of numbers f(n,k) from n-th differences of sequence {1/x^2}; n-th difference is n!*P(x)/(D^2) where P(x) is a degree-n polynomial: P(n) = Sum_k { f(n,k)*x^k } and D = x(x+1) ...(x+n-1)(x+n).

A034855 0 LD Triangle read by rows giving number of rooted labeled trees with n >= 2 nodes and height d >= 1.

A034893 0 SD Maximum of different products of partitions of n into distinct parts.

A036970 0 LD Triangle of coefficients of Gandhi polynomials.

A038719 0 LD Triangle T(n,k) (0<=k<=n) giving number of chains of length k in partially ordered set formed from subsets of n-set by inclusion.

A042977 0 LD Triangle T(n,k) read by rows: coefficients of a polynomial sequence occurring when calculating the n-th derivative of Lambert function W.

A046900 0 LD Triangle inverse to that in A046899.

A047887 0 LD Triangle of numbers T(n,k) = number of permutations of n things with longest increasing subsequence of length <=k (1<=k<=n).

A047916 0 LD Triangular array read by rows: a(n,k) = phi(n/k)*(n/k)^k*k! if k|n else 0 (1<=k<=n).

A047917 0 LD Triangular array read by rows: a(n,k) = phi(n/k)*(n/k)^k*k!/n if k|n else 0 (1<=k<=n).

A047920 0 FC Triangular array formed from successive differences of factorial numbers.

A047922 0 FC Triangle of numbers a(n,k) = number of terms in n X n determinant with 2 adjacent diagonals of k and k-1 0's (0<=k<=n).

A047991 0 LD Triangle formed from expansion of (x-1)(x+2)(x-3)...(x+-n).

A048594 0 FC Triangle a(n,k) = k! * Stirling1(n,k), 1<=k<=n.

A048594 0 LD Triangle a(n,k) = k! * Stirling1(n,k), 1<=k<=n.

A048743 0 LD Triangle a(n,k) = k!*C(n-1,k-1)*Stirling_2(n,k), 1<=k<=n.

A048994 0 SC Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0<=k<=n.

A048998 0 LD Triangle giving coefficients of (n+1)!*B_n(x), where B_n(x) is a Bernoulli polynomial. Rising powers of x.

A048999 0 FC Triangle giving coefficients of (n+1)!*B_n(x), where B_n(x) is a Bernoulli polynomial, ordered by falling powers of x.

A049055 0 LD Triangle read by rows, giving T(n,k) = number of k-member minimal ordered covers of a labeled n-set (1 <= k <= n).

A049444 0 FC Generalized Stirling number triangle of first kind.

A051683 0 FC Triangle read by rows: a(n,k)=n!*k.

A053440 0 LD Number of k-simplices in the first derived complex of the standard triangulation of an n-simplex. Equivalently, T(n,k) is the number of ascending chains of length k+1 of nonempty subsets of the set {1, 2, ..., n+1}.

A053495 0 LD Triangle formed by coefficients of numerator polynomials defined by iterating f(u,v) = 1/u - x*v applied to a list of elements {1,2,3,4,...}.

A053495 0 SD Triangle formed by coefficients of numerator polynomials defined by iterating f(u,v) = 1/u - x*v applied to a list of elements {1,2,3,4,...}.

A054115 0 SC Triangular array generated by its row sums: T(n,0)=1 for n >= 1, T(n,1)=r(n-1), T(n,k)=T(n,k-1)+r(n-k) for k=2,3,...,n, n >= 2, r(h)=sum of the numbers in row h of T.

A054255 0 FC Triangle T(n,k) (n >= 1, 0<=k<=n) giving number of preferential arrangements of n things beginning with k (transposed, then read by rows).

A054589 0 FC Table related to labeled rooted trees, cycles and binary trees.

A054651 0 LD Triangle T(n,k) read by rows giving coefficients in expansion of n! * Sum_{i=0..n} C(x,i) in descending powers of x.

A054654 0 SD Triangle read by rows: matrix product of the binomial coefficients with the Stirling numbers of the first kind.

A055302 0 FC Triangle of labeled rooted trees with n nodes and k leaves, n>=1, 1<=k<=n.

A055349 0 FC Triangle of labeled mobiles (circular rooted trees) with n nodes and k leaves.

A055356 0 SD Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.

A055924 0 FC Exponential transform of Stirling-1 triangle A008275.

A055925 0 FC Exponential reciprocal of A055924.

A056151 0 LD Distribution of maximum inversion table entry.

A056856 0 FC Triangle of numbers related to rooted trees and unrooted planar trees.

A058298 0 LD Triangle n!/(n-k), 1 <= k < n, read by rows.

A058942 0 LD Triangle of coefficients of Gandhi polynomials.

A059098 0 LD Triangle T(n,m) = Sum_{i=0..n} stirling2(n,i)*Product_{j=1..m} (i-j+1), m=0..n.

A059114 0 LD Triangle T(n,m)= Sum_{i=0..n} L'(n,i)*Product_{j=1..m} (i-j+1), m=0..n.

A059286 0 SD Triangle T(n,k) (0 <= k <= n) with e.g.f. exp(x*y/(1-y))

A059364 0 FC Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.

A059364 0 LD Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.

A059369 0 LD Triangle of numbers T(n,k) = T(n-1,k-1) + ((n+k-1)/k)*T(n-1,k), n >= 1, 1<=k<=n, with T(n,1) = n!, T(n,n) = 1; read from right to left.

A059374 0 LD Triangle T(n,k)=Sum_{i=0..n} L'(n,n-i)*binomial(i,k), k=0..n-1.

A059446 0 FC Triangle T(n,k) = coefficient of x^n*y^k/(n!*k!) in 1/(1-x-y-x*y), read by rows in order 00, 10, 01, 20, 11, 02, ...

A059446 0 LD Triangle T(n,k) = coefficient of x^n*y^k/(n!*k!) in 1/(1-x-y-x*y), read by rows in order 00, 10, 01, 20, 11, 02, ...

A060538 0 LD Square array read by antidiagonals of number of ways of dividing nk labeled items into n labeled boxes with k items in each box.

A060637 0 SC Triangle T(n,k) (0 <= k <= n) giving number of tilings of the unary zonotope Z(n,k) (the projection onto R^k of a unit cube in R^n) by projections of the k-dimensional faces of the hypercube (again projected onto R^k).

A060694 0 LD A triangle related to rooted trees.

A061018 0 FC Triangle: a(n,m) = number of permutations of (1,2,...,n) with one or more fixed points in the m first positions.

A061691 0 LD Triangle of generalized Stirling numbers.

A066121 0 SC Multi-level factorials: triangle with a(n,k)=a(n-1,k-1)*a(n-1,k) but with a(n,1)=n and a(n,n)=1.

A066324 0 LD Number of endofunctions on n labeled points constructed from k rooted trees.

A066387 0 LD Triangle T(n,m) (1<=m<=n) giving number of maps f:N -> N such that f^m(X)=X+n for all natural numbers X.

A066667 0 FC Coefficient triangle of generalized Laguerre polynomials (a=1).

A066991 0 LD Square array read by antidiagonals of number of ways of dividing nk labeled items into k unlabeled orders with n items in each order.

A067050 0 SD Triangle T(n,r), n>=0, r=n, n-1, ..., 1, 0; where T(n,r) = product of all possible sums of r numbers chosen from [1..n].

A067948 0 FC Triangle of labeled rooted trees according to the number of increasing edges.

A067948 0 LD Triangle of labeled rooted trees according to the number of increasing edges.

A068106 0 LD Euler's difference table: triangle read by rows, formed by starting with factorial numbers (A000142) and repeatedly taking differences. T(n,n) = n!, T(n,k) = T(n,k+1)-T(n-1,k).

A068424 0 LD Triangle of falling factorials, read by rows: T(n, k) = n(n-1)...(n-k+1), n > 0, 1 <= k <= n.

A068424 0 SD Triangle of falling factorials, read by rows: T(n, k) = n(n-1)...(n-k+1), n > 0, 1 <= k <= n.

A069777 0 SC Triangle of q-factorial numbers n!_q, for (n,q) = (0,0), (1,0), (0,1), (2,0), etc.

A071208 0 FC Triangular array T(n,k) read by rows, giving number of labeled free trees such that the root is smaller than all its children, with respect to the number n of vertices and to the number k of decreasing edges.

A071208 0 LD Triangular array T(n,k) read by rows, giving number of labeled free trees such that the root is smaller than all its children, with respect to the number n of vertices and to the number k of decreasing edges.

A071223 0 LD Triangle T(n,k) (n >= 2, 1 <= k <= n) read by rows: number of linearly inducible orderings of n points in k-dimensional Euclidean space.

A073315 0 FC Expansion of Lambert W function in powers of log(log(x))/log(x).

A073315 0 LD Expansion of Lambert W function in powers of log(log(x))/log(x).

A073474 0 LD Triangle T(n,k) read by rows, where o.g.f. for T(n,k) is n!*Sum_{k=0..n} (1+x)^(n-k)/k!.

A073474 0 SD Triangle T(n,k) read by rows, where o.g.f. for T(n,k) is n!*Sum_{k=0..n} (1+x)^(n-k)/k!.

A073768 0 FC Triangle of coefficients of Bateman polynomial n!Z_n(-x).

A074911 0 FC Triangle generated by Pascal's rule, except begin and end the n-th row with n!.

A074911 0 LD Triangle generated by Pascal's rule, except begin and end the n-th row with n!.

A075181 0 FC Coefficients of certain polynomials (rising powers).

A075181 0 LD Coefficients of certain polynomials (rising powers).

A075263 0 FC Triangle of coefficients of polynomials H(n,x) formed from the first (n+1) terms of the power series expansion of ( -x/log(1-x) )^(n+1), multiplied by n!.

A075856 0 FC Triangle formed from coefficients of the polynomials p(1)=x, p(n+1)=(n+x*(n+1))*p(n)+x*x*diff(p(n),x).

A076256 0 LD Coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the constant term.

A076257 0 FC Coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the coefficient of the highest power of x.

A076571 0 FC Binomial triangle based on factorials.

A077012 0 FC Triangle in which n-th row contains all possible products of n-1 of the first n natural numbers in ascending order.

A077012 0 LD Triangle in which n-th row contains all possible products of n-1 of the first n natural numbers in ascending order.

A078341 0 LD Triangle read by rows: T(n,k) = n*T(n-1,k-1)+k*T(n-1,k) starting with T(0,0)=1.

A078921 0 FC Signed variant of A077012.

A078921 0 LD Signed variant of A077012.

A078991 0 LD Coefficients of the polynomials in the numerator of the generating function x/(1-x-x^2) for the Fibonacci sequence and its successive derivatives starting with the highest power of x.

A078992 0 FC Nonzero coefficients of the polynomials in the numerator of the generating function x/(1-x-x^2) for the Fibonacci sequence and its successive derivatives starting with the highest power of x.

A079005 0 FC Exponential transform of unsigned Lah-triangle |A008297(n,k)|.

A079461 0 SD Coefficients of the polynomials in the numerator of the generating function x/(1-x-x^2) for the Fibonacci sequence and its successive derivatives starting with the constant.

A079462 0 LD Nonzero coefficients of the polynomials in the numerator of the generating function x/(1-x-x^2) for the Fibonacci sequence and its successive derivatives starting with the constant.

A079510 0 LD Triangle T(n,k) read by rows; related to number of preorders.

A080779 0 LD Triangle read by rows: n-th row gives expansion of the series for HarmonicNumber[n, -p].

A080955 0 LD Square array of numbers related to the incomplete gamma function, read by antidiagonals.

A081957 0 FC In the following triangle the n-th row contains n smallest numbers that are products of n distinct integers >1. Sequence contains the triangle by rows.

A082037 0 LD A square array of linear-factorial numbers, read by antidiagonals.

A082037 0 SD A square array of linear-factorial numbers, read by antidiagonals.

A082038 0 LD A square array of quadratic-factorial numbers, read by antidiagonals.

A084416 0 LD Triangle read by rows: T(n,k) = Sum_{i=k..n} i!*Stirling2(n,i), n >= 1, 1 <= k <= n.

A084417 0 FC Triangle read by rows: T(n,k)=sum((n+1-i)!*stirling2(n,n+1-i),i=1..k), n>=1, 1<=k<=n.

A084938 0 SC Triangle read by rows: T(n,k) = Sum_{j>=0} j!*T(n-j-1, k-1) for n >= 0, k >= 0.

A084980 0 LD Triangle of (multi)factorials: n-th row is (n+1)!!... {n "!"s}, (n+1)!... {n-1 "!"s}, ..., (n+1)!.

A087854 0 LD Triangle read by rows: T(n,k) is the number of n-bead necklaces with exactly k different colored beads.

A088996 0 LD Triangle T(n,k) read by rows, given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938.

A088996 0 SC Triangle T(n,k) read by rows, given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938.

A089087 0 FC Triangular array of coefficients multiplied by n! of polynomials in e. These give the expected number of trials needed for the sum of uniform random variables from the interval [0,1] to exceeds n+1.

A089231 0 LD Triangular array A066667 or A008297 unsigned and transposed.

A089258 0 SD Another version of A080955.

A089759 0 SC Table T(n,k), 0<=k, 0<=n, read by antidiagonals, defined by T(n,k) = (k*n)! / (n!)^k.

A089900 0 FC Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the factorials, starting with row 0: {1!,2!,3!,...}.

A089949 0 LD Triangle T(n,k), read by rows, given by : [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938.

A090238 0 SC Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

A090441 0 SC Symmetric triangle of certain normalized products of decreasing factorials.

A090441 0 SD Symmetric triangle of certain normalized products of decreasing factorials.

A090582 0 FC Numerator Q(m,n) of probability P(m,n)=Q(m,n)/n^m of seeing each card at least once if m>=n cards are drawn with replacement from a deck of n cards, written in a two-dimensional array read by antidiagonals with Q(m,m) as first row and Q(m,1) as first column.

A090628 0 SD Square array T(n,k) (row n, column k) read by antidiagonals defined by: T(n,k) is the permanent of the n X n matrix with 1 on the diagonal and k elsewhere; T(0,k)=1.

A090657 0 LD Triangle read by rows: T(n,k) = number of functions from [1,2,...,n] to [1,2,...,n] such that the image contains exactly k elements (0<=k<=n).

A090665 0 LD Triangle read by rows: T(n,k) = number of preferential arrangements of n things beginning with k.

A090802 0 LD Triangle read by rows: a(n,k) = number of k-length walks in the Hasse diagram of a Boolean algebra of order n.

A091441 0 FC Table (by antidiagonals) of permutations of two types of objects so that each cycle contains at least one object of each type. Each type of object labeled from its own label set.

A091441 0 LD Table (by antidiagonals) of permutations of two types of objects so that each cycle contains at least one object of each type. Each type of object labeled from its own label set.

A092271 0 FC First in a series of triangular arrays counting permutations of partitions.

A093447 0 FC Triangle a(n,k) read by rows n which contain columns k=1,2,..,n, where each entry is the product of numbers (k-1)*n-T(k-2)+1 through k*n-T(k-1).

A094310 0 FC Triangle read by rows: T(n,k), the k-th term of the n-th row, is the product of all numbers from 1 to n except k: T(n,k) = n!/k.

A094310 0 LD Triangle read by rows: T(n,k), the k-th term of the n-th row, is the product of all numbers from 1 to n except k: T(n,k) = n!/k.

A094344 0 SC Triangle T(n,k), 0<= k <= n, read by rows; given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ...] DELTA [1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...] where DELTA is the operator defined in A084938.

A094346 0 LD Another version of triangular array in A036970: triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ...] where DELTA is the operator defined in A084938.

A094485 0 FC T(n,k) = Stirling1(n+1,k+1)-Stirling1(n,k), n>=1, 0<=k<=n-1.

A094587 0 FC Triangle of permutation coefficients arranged with 1's on the diagonal. Also, triangle of permutations on n letters with exactly k+1 cycles and with the first k+1 letters in separate cycles.

A094587 0 SC Triangle of permutation coefficients arranged with 1's on the diagonal. Also, triangle of permutations on n letters with exactly k+1 cycles and with the first k+1 letters in separate cycles.

A094638 0 LD Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind (1<=k<=n; in other words, the unsigned Stirling numbers of the first kind in reverse order).

A094645 0 SC Generalized Stirling number triangle of first kind.

A096747 0 LD Triangle read by rows: T(n,1) = 1, T(n,k) = T(n-1,k)+(n-1)T(n-1,k-1) for 1<=k<=n+1.

A096747 0 SD Triangle read by rows: T(n,1) = 1, T(n,k) = T(n-1,k)+(n-1)T(n-1,k-1) for 1<=k<=n+1.

A097905 0 FC Triangle where a(m,n) = largest divisor of m! coprime to n.

A098361 0 FC Multiplication table of the factorial numbers read by antidiagonals.

A098361 0 LD Multiplication table of the factorial numbers read by antidiagonals.

A098361 0 SC Multiplication table of the factorial numbers read by antidiagonals.

A098361 0 SD Multiplication table of the factorial numbers read by antidiagonals.

A099394 0 LD Triangle T(k,n) by rows: n! * A075499(k,n).

A099599 0 LD Triangle T read by rows: coefficients of polynomials generating array A099597.

A100630 0 FC Array read by antidiagonals: T(m,n) = Sum(1<=i<=m) [ i*(n-1+i)! ]

A100822 0 FC Triangle read by rows: T(n,k) is the number of deco polyominoes of height n with k cells in the first column. (A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column).

A101032 0 LD Table (read by rows) giving the coefficients of sum formulas of n-th Lucas numbers (A000204). The k-th row (k>=1) contains T(i,k) for i=1 to k, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies L(n) = Sum_{i=1..k} T(i,k) * n^(k-i) / (k-1)!.

A101817 0 LD Triangle read by rows: T(n,h) = number of functions f:{1,2,...,n}->{1,2,...,n} such that |Image(f)|=h; h=1,2,...,n, n=1,2,3,... . Essentially A090657, but without zeros.

A101818 0 LD Triangle read by rows: (1/n)*T(n,h), where T(n,h) is the array in A101817.

A102625 0 LD Triangle read by rows: T(n,k) is the sum of the weights of all vertices labeled k at depth n in the Catalan tree (1<=k<=n+1, n>=0).

A103880 0 FC Square array T(n,k) read by antidiagonals: denominators of Stirling numbers of first kind with negative argument S1(-n,k), n,k>=0.

A104001 0 LD Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns starting with fixed m, 2<k<=n, 1<=m<=k.

A104035 0 LD Triangle T(n,k), 0<=k<=n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k>0 or if k<0; T(n,k) = k*T(n-1,k-1) + (k+1)*T(n-1,k+1).

A104557 0 FC Triangle T, read by rows, such that the unsigned columns of the matrix inverse when read downwards equals the rows of T read backwards, with T(n,n)=1 and T(n,n-1) = [(n+1)/2]*[(n+2)/2].

A104557 0 SC Triangle T, read by rows, such that the unsigned columns of the matrix inverse when read downwards equals the rows of T read backwards, with T(n,n)=1 and T(n,n-1) = [(n+1)/2]*[(n+2)/2].

A104601 0 LD Triangle T(r,n) read by rows: number of n X n (0,1)-matrices with exactly r entries equal to 1 and no zero row or columns.

A105196 0 FC Triangle, read by rows, of Stirling numbers of first kind, S1(n,k), multiplied by k^k, for n >= 1, 1<=k<=n.

A105278 0 FC Triangle read by rows: T(n,k) = C(n,k)*(n-1)!/(k-1)!.

A105725 0 FC Triangle read by rows: T(n,k)=(n+k)!/k! (0<=k<=n-1; n>=1).

A105725 0 SC Triangle read by rows: T(n,k)=(n+k)!/k! (0<=k<=n-1; n>=1).

A105793 0 SC Expansion of (1+y)^(1+x).

A105954 0 LD Array read by antidiagonals: a(m,n) = m!*H(n,m), where H(n,m) is a higher-order harmonic number (H(0,m) = 1/m; H(n,m) = Sum_{k=1..m} H(n-1,k)).

A106338 0 LD Triangle T, read by rows, equal to the matrix inverse of the triangle defined by [T^-1](n,k) = A075263(n,k)/n!, for n>=k>=0.

A107729 0 LD Triangle T(n,k), 0<=k<=n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k<0 or if k>0; T(n,k) = k*T(n-1,k-1) + (k+2)*T(n-1,k+1).

A107893 0 LD Triangle read by rows, related to A055129 (repunits in base k).

A108032 0 LD Triangle T(n,k), 0<=k<=n, read by rows, defined by : T(0,0) = 1, T((n,k) = 0 if n<k or if k<0, T(n,k) = k*T(n-1, k-1) + (2n-2k-1)*T(n-1, k).

A108032 0 SD Triangle T(n,k), 0<=k<=n, read by rows, defined by : T(0,0) = 1, T((n,k) = 0 if n<k or if k<0, T(n,k) = k*T(n-1, k-1) + (2n-2k-1)*T(n-1, k).

A108284 0 LD Triangle read by rows, related to A108283.

A109822 0 LD Triangle read by rows: T(n,1)=1, T(n,k)=T(n-1,k)+(n-1)T(n-1,k-1) for 1<=k<=n.

A109876 0 LD Triangle read by rows: a(n, n) = n! and for 1 <= k < n, a(n, k) = sum_{i=0..n-1} prod_{j=i+1..i+k} f(j, n), where for x <= y, f(x, y) = x and for x > y, f(x, y) = x-y.

A109878 0 FC Triangle read by rows: see below.

A109878 0 SC Triangle read by rows: see below.

A110768 0 LD The r-th term of the n-th row of the following triangle contains product of r successive numbers in decreasing order beginning from T(n)-T(r-1) where T(n) is the n-th triangular number. 1 3 2 6 20 6 10 72 210 24 15 182 1320 3024 120 ... Sequence contains the triangle by rows.

A111184 0 SC Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938.

A111467 0 FC The following triangle contains n smallest numbers with the prime signature of n!. Sequence contains the triangle by rows.

A111492 0 LD Triangle read by rows: a(n,k) = (k-1)! * C(n,k).

A111528 0 LD Square table, read by antidiagonals, where the g.f. for row n+1 is generated by: R_{n+1}(x) = (1+n*x - 1/R_n(x))/(n+1) with R_0(x) = Sum_{n>=0} n!*x^n.

A111596 0 SC Associated Sheffer triangle to Sheffer triangle A111595.

A111999 0 LD A triangle that converts certain binomials into triangle A008276 (diagonals of signed Stirling1 triangle A008275).

A112007 0 FC Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.

A112356 0 SC Following triangle is based on Pascal's triangle. The r-th term of the n-th row is product of C(n,r) successive integers such that the product of all the terms of the row is (2^n)!. Sequence contains the triangle read by rows.

A112486 0 FC Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.

A113129 0 LD Triangle T(n,k), 0<=k<=n, of coefficients of polynomials P_n(x) related to convolution of the k-fold factorials.

A114423 0 FC Multifactorial array read by antidiagonals.

A116853 0 LD Difference triangle of factorial numbers read by upward diagonals.

A116854 0 LD Triangle T(n,k) = A116853(n,k) - A116853(n,k-1) read by rows.

A116923 0 LD Triangle T(n,k) = Sum_{i=0..k} (-1)^(i+k)*binomial(k,i)*Sum_{j=0..n} (i+1)^j*(3n-3j+1) read by rows.

A118787 0 FC Triangle where T(n,k) = n!*[x^k] ( x/(2*x + log(1-x)) )^(n+1), for n>=k>=0, read by rows.

A118791 0 FC Triangle where T(n,k) = -n!*[x^k] ( x/log(1-x-x^2) )^(n+1), for n>=k>=0, read by rows.

A118984 0 LD Triangular T(n,k) which contains in column k>=0 the elements of the Stirling transform of the unsigned sequence Stirling1(j+k,j), j>=0.

A119502 0 FC Triangle read by rows, T(n,k) = (n-k)!, for n>=0 and 0<=k<=n.

A119502 0 SC Triangle read by rows, T(n,k) = (n-k)!, for n>=0 and 0<=k<=n.

A119741 0 LD A008279, with the first and last of each row removed.

A121757 0 LD Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.

A122525 0 FC Triangle read by rows: G(s,rho) = ((s-1)!/s)*Sum(((s-i)/i!)*(s*rho)^i, i=0..(s-1)).

A123202 0 FC Triangular array formed from the coefficients of the Bezier transform of A021009(n,m): t(n,m,x)=A021009(n,m)*x^m*(1 - x)^(n - m).

A123316 0 FC Triangle read by rows: T(n,k)=(k+1)*n!/2 (1<=k<=n).

A123361 0 FC Triangle read by rows: T(n,k)=coefficient of x^k in the polynomial p[n,x] defined by p[0,x]=1, p[1,x]=1+x and p[n,x]=(1+x)(2-x)(3-x)...(n-x) for n>=2 (0<=k<=n).

A123670 0 LD Triangle read by rows: T(n,k) is the coefficient of x^k of the polynomial n(n-x)(n-2x)(n-3x)...(n-(n-1)x) (n>=1, 0<=k<=n-1).

A125714 0 FC Alfred Moessner's factorial triangle.

A126064 0 FC Triangle read by rows, obtained by multiplying columns of triangle in A094587 by 1,2,4,8,16,... respectively.

A126074 0 LD Triangle read by rows: T(n,k) is the number of permutations of n elements that have the longest cycle length k.

A126671 0 SC Triangle read by rows: row n (n>=0) has g.f. Sum_{i=1..n} n!*x^i*(1+x)^(n-i)/(n+1-i).

A127755 0 LD Inverse of number triangle A(n,k)=if(k<=n,if(n<=2k,1/n!,0),0).

A127755 0 SD Inverse of number triangle A(n,k)=if(k<=n,if(n<=2k,1/n!,0),0).

A128264 0 FC Triangle read by rows, 1<=m<=n: t(n,m) = LCM(s(n,m),S(n,m)), where s(n,m) is an unsigned Stirling number of the first kind and S(n,m) is a Stirling number of the second kind.

A129116 0 LD Multifactorial array A[k,n] = k-tuple factorial of n, for positive n, by antidiagonals.

A130469 0 FC Triangular array read by rows: T(j,k) = k*(j-k)! for k < j, T(j,k) = 1 for k = j; 1 <= k <= j.

A130478 0 FC Triangle T(n,k) = n! / A130477(n,k).

A130493 0 FC Triangle read by rows in which row n contains n! repeated n times.

A130493 0 LD Triangle read by rows in which row n contains n! repeated n times.

A130493 0 SC Triangle read by rows in which row n contains n! repeated n times.

A130493 0 SD Triangle read by rows in which row n contains n! repeated n times.

A130534 0 FC Triangle T(n,k), 0<=k<=n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1,k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.

A130562 0 LD Triangular table of denominators of the coefficients of Laguerre-Sonin polynomials L(1/2,n,x).

A130850 0 FC Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.

A131182 0 SC Table T(n,k)=n!*k^n, read by antidiagonals .

A131689 0 LD Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows (n >= 0, 0 <= k <= n).

A131758 0 LD Coefficients of numerators of rational functions whose binomial transforms give the normalized polylogarithms Li(-n,t)/ n!.

A132159 0 FC Lower triangular matrix T(n,j) for double application of an iterated mixed order Laguerre transform inverse to A132014. Coefficients of Laguerre polynomials (-1)^n * n! * L(n,-2-n,x).

A132393 0 SC Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows.

A132986 0 SD Triangle, read by rows, where row n of T = row n-1 of T^n (shift right 1 column) with T(n,0)=T(n,1) for n>0.

A133643 0 LD Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = smallest permanent of any n X n (0,1) matrix with k 1's in each row and column.

A133644 0 LD Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = largest permanent of any n X n (0,1) matrix with k 1's in each row and column.

A134134 0 FC Triangle of numbers obtained from the partition array A134133.

A134433 0 SC Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which the last entry of the first increasing run is equal to k (1<=k<=n).

A134436 0 LD Triangle read by rows: T(n,k) is the number of deco polyominoes of height n with k cells in the second row (0<=k<=n-1; a deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column).

A134558 0 FC Array, a(n,k) = Gamma[n+1,k]*e^k, where Gamma[n,k] is the upper incomplete gamma function and e is the exponential constant 2.71828..., read by anti-diagonals...

A134830 0 FC Triangle of rank k of permutations of {1,2,...,n}.

A135338 0 FC Triangle read by rows: row n gives coefficients C(n,j) for a Sheffer sequence (binomial-type) with raising operator -x { 1 + W[ -exp(-2) * (2+D) ] } where W is the Lambert W multi-valued function.

A136124 0 FC Triangle read by rows: T(n,k)=(-1)^(n+k)*Sum(s(n,j),j=1..k), where s(n,j) are the signed Stirling numbers of the first kind (n>=2; 1<=k<=n-1; s(n,j)=A008275(n,j)).

A136125 0 LD Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which the size of the last cycle is k (the cycles are ordered by increasing smallest elements; 1 <= k <=n).

A136125 0 SD Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which the size of the last cycle is k (the cycles are ordered by increasing smallest elements; 1 <= k <=n).

A136426 0 SC Triangle T(n,0)=0 and T(n,k) = -A028421(n-1,k-1), 0<k<=n.

A136572 0 LD Triangle read by rows: row n consists of n zeros followed by n!.

A136573 0 FC Triangle read by rows: (A000012 * A136572 + A136572 * A000012) - A000012.

A136573 0 SC Triangle read by rows: (A000012 * A136572 + A136572 * A000012) - A000012.

A136579 0 LD Triangle read by rows: A128174 * A136572.

A136581 0 LD Triangle read by rows: A136572 * A128174.

A136656 0 SC Coefficients for rewriting generalized falling factorials into ordinary falling factorials.

A137216 0 FC Erlang C queues type triangular sequence based on A122525.

A137227 0 FC A triangular sequence of queues like A122525 that depends on a Fibonacci sequence in a differential way by replacing: m->A000045[m+1]-A000045[m].

A137267 0 LD Chung-Graham juggling polynomials as a triangular sequence of positive coefficients.

A137268 0 LD Period and balls triangular sequence for Juggling from Chung-Graham.

A137268 0 SD Period and balls triangular sequence for Juggling from Chung-Graham.

A137376 0 FC Triangular sequence from coefficients of Meixner polynomials from expansion of: p(x)=(1 - t/c)*(1 - t)^(-x - b);c = 1/2; b = 1;.

A137381 0 LD Triangular sequence of coefficients from expansion of Narumi polynomials: generated by:p(x) = (t/Log[1 + t])^a0*(1 + t)^x; a0=2;weights (n+1)!*n!;.

A137387 0 SC Triangular sequence from coefficients of the expansion of p(x,t)=Exp[2*x*t]*t/(1 - t).

A137394 0 FC Triangular sequence from a Pidduck polynomials expansion: p(t) = (t/(1 - t))*((1 + t)/(1 - t))^x.

A137477 0 LD A triangular sequence of coefficients from the inverse substitution of in the spherical Bessel polynomial recursion:k=1;x->1/y; B(x, n) = (-2/x)*B(x, n - 1) - (k^2 - (n*(n - 1)/x^2))*B(x, n - 2).

A137524 0 LD Triangular sequence from coefficients of the umbral calculus expansion of a Golden -Mean Bernoulli function(A001898): p(x,t)=t*phi^(x*t)/(phi^t - 1), where the golden ratio replaces "e".

A137525 2 SD A triangular sequence of coefficients based on an expansion of a Catenoid like Sheffer expansion function: g(t) = t; f(t) = -1/t; p(x,t) = Exp[x*(t)]*(1 - f(t)^2).

A137948 0 LD Triangle read by rows, A000012 * A136579.

A138024 0 FC A triangular sequence of coefficients of an expansion of a Mach wave as a traveling wave in a medium: (vt')^2=vp*vg=c^2-(gamma-1)/gamma+1)*vt^2; Substituting: vt-> Exp[t*x];gamma->t;c->1; p(x,t)=1 - Exp[2*x*t]*(t - 1)/(1 + t).

A138133 3 SD Triangle read by rows, based on the two-variable g.f. exp(x*t)*(x*(1 - 2*exp(x)) - 2*exp(x))/(1 - exp(t)) (the first of two parts).

A138771 0 FC Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} whose 2nd cycle has k entries; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements (n>=1; 0<=k<=n-1). For example, 1432=(1)(24)(3) has 2 entries in the 2nd cycle; 3421=(1324) has 0 entries in the 2nd cycle.

A138771 0 LD Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} whose 2nd cycle has k entries; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements (n>=1; 0<=k<=n-1). For example, 1432=(1)(24)(3) has 2 entries in the 2nd cycle; 3421=(1324) has 0 entries in the 2nd cycle.

A139359 0 LD Number L([n],m) of ways the labelled parts of each integer partition of n can be distributed into m nonempty labelled boxes.

A140333 0 LD Triangle T(n,k) with the coefficient [x^k] (n+1)!* C(n,x), in row n, column k, where C(.,.) are the Bernoulli twin number polynomials of A129378.

A140713 0 LD Triangle read by rows: T(n,k) is the number of white corners of rank k in all the permutations of {1,2,...,n} (n>=2, 0<=k<=n-2; for definitions see the Eriksson-Linusson references).

A140956 0 FC Triangle read by rows: coefficients of the alternating factorial polynomial (x+1)(x-2)(x+3)(x-4)...(x+n*(-1)^(n-1)).

A141476 0 FC Triangle T(n,k) = A000142(n-k)*A003319(k+1) read by rows.

A141476 0 SC Triangle T(n,k) = A000142(n-k)*A003319(k+1) read by rows.

A141618 0 LD Triangle read by rows: number of nilpotent partial transformations (of an n-element set) of height r (height(alpha) = |Im(alpha)|), 0<=r<n.

A141906 0 SC Triangle t(n,m) = (n*m)!/(m!^n) read by rows, 0<=m<=n.

A142070 0 FC A triangle of coefficients of rational root polynomials: p(x,n)=Product[(i + 1)*x - i, {i, 1, n}].

A142070 0 LD A triangle of coefficients of rational root polynomials: p(x,n)=Product[(i + 1)*x - i, {i, 1, n}].

A142071 0 SD A triangle sequence of coefficients of an infinite sum polynomial: p(x,n)=Sum[k^n*(x/(1 + x))^k, {k, 0, Infinity}]=PolyLog[ -n,x/(1+x)].

A142148 0 FC A triangular sequence of polynomial coefficients of an adjusted root product one polynomial set: w(i,n)=If[i == 1, 1/n!, i]; p(x,n)=n!*Product[x - w[i, n], {i, 0, n}]/x.

A142148 0 LD A triangular sequence of polynomial coefficients of an adjusted root product one polynomial set: w(i,n)=If[i == 1, 1/n!, i]; p(x,n)=n!*Product[x - w[i, n], {i, 0, n}]/x.

A142156 0 FC Triangle T(n,k)= n! if k=0, T(n,k) = -(n-k)!*A003319(k) if k>0.

A142156 0 SC Triangle T(n,k)= n! if k=0, T(n,k) = -(n-k)!*A003319(k) if k>0.

A142336 0 LD A generalized PolyLog triangular sequence of coefficients: k = (n + 1); b0 = 1; p(x,n,k)=(k - 1)!*(1 - x)^n*PolyLog[ -n, k, x]/(x*Log[1 - x]); t(n,m)=Coefficients(p(b0,n,k)).

A142473 0 LD A division triangle sequence of the Stirling numbers of the first kind by the binomial ( Pascal's triangle): t(n,m)=n!*StirlingS1[n, m]/Binomial[n, m].

A142589 0 SC Square array A(n,m) = product_{i=0..m} (1+n*i) read by antidiagonals.

A143084 0 FC Triangle sequence: t(n,m)=(n+m)!.

A143084 0 SC Triangle sequence: t(n,m)=(n+m)!.

A143085 0 FC Triangle sequence: t(n,m)=(n+1)*(n+m)!.

A143216 0 FC Triangle read by rows, T(n,k) = n!*k!, 0<=k<=n.

A143216 0 SC Triangle read by rows, T(n,k) = n!*k!, 0<=k<=n.

A143491 0 FC Unsigned 2-restricted Stirling numbers of the first kind.

A143806 0 FC Eigentriangle of A130534

A143965 0 FC Factorial eigentriangle: A119502 * (A051295 *0^(n-k); 0<=k<=n

A143965 0 SC Factorial eigentriangle: A119502 * (A051295 *0^(n-k); 0<=k<=n

A144084 0 LD T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.

A144107 0 LD Eigentriangle, row sums = n!

A144107 0 SD Eigentriangle, row sums = n!

A144108 0 LD Eigentriangle based on A052186 (permutations without strong fixed points), row sums = n!

A144351 0 FC Lower triangular array called S1hat(1) related to partition number array A107106.

A145141 0 LD Denominators of triangle T(n,k), n>=1, 0<=k<=n - 1, read by rows: T(n,k) is the coefficient of x^k in polynomial p_n for the n-th row sequence of A145153.

A145324 0 LD Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....

A145877 0 LD Triangle read by rows: T(n,k) is the number of permutations of [n] for which the shortest cycle length is k (1<=k<=n).

A145888 0 FC Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which k is the largest entry in the cycle containing 1 (1<=k<=n).

A145888 0 SC Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which k is the largest entry in the cycle containing 1 (1<=k<=n).

A152656 0 FC Triangle read by rows: denominators of polynomials from A000142: P(0,x)=1,P(n,x)=1/n!+x*sum(P(n-i-1)/i!),i=0,1,..,n-1. Numerators are A152650.

A152818 0 LD Array read by antidiagonals: T(n,k) = (k+1)^n*(n+k)!/n!.

A152883 0 LD Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which k is an excedance (n>=2, 1<=k<=n-1). An excedance of a permutation p is a value j such p(j)>j.

A152918 0 FC Triangle read by rows based on the Stirling numbers S1: t(n,m)=Sum[(-1)^(n + 1)* StirlingS1[n, j]*(k + 1 - j)^(n - 1), {j, 0, k + 1}].

A153274 0 FC A Pochhammer function-based triangular sequence: w(n,m,j)=m^(n + 1)*Pochhammer[j/m,n+1]; t(n,m)=sum_coefficients(w(n,m,j) in j).

A154120 0 FC Array read by antidiagonals: T(n,k) = (k+1)^n*(n+k)!.

A154120 0 LD Array read by antidiagonals: T(n,k) = (k+1)^n*(n+k)!.

A155100 0 LD Triangle read by rows: coefficients in polynomials P_n(u) arising from the expansion of D^(n-1) (tan x) in increasing powers of tan x for n>=1 and 1 for n=0.

A155453 0 FC Triangle read by rows: t(n,m)=1 - (k! - n! + (-k + n)! - k!*(n - k)!).

A155453 0 LD Triangle read by rows: t(n,m)=1 - (k! - n! + (-k + n)! - k!*(n - k)!).

A155453 0 SC Triangle read by rows: t(n,m)=1 - (k! - n! + (-k + n)! - k!*(n - k)!).

A155453 0 SD Triangle read by rows: t(n,m)=1 - (k! - n! + (-k + n)! - k!*(n - k)!).

A155794 0 SC Triangle read by rows: t(n,m)=(m*(m-n))!

A155794 0 SD Triangle read by rows: t(n,m)=(m*(m-n))!

A155795 0 SC Triangle read by rows: t(n,k)=n!/(n - k*(n - k)).

A155795 0 SD Triangle read by rows: t(n,k)=n!/(n - k*(n - k)).

A155856 0 FC Triangle T(n,k)=C(2n-k,k)*(n-k)!

A156367 0 LD Number triangle T(n,k)=C(n+k,2k)*k!

A156528 0 FC Triangle read by rows: T(n,m) = (-1)^n*Sum_{i=0..m} [(-1)^(m-i)*binomial(n- i-1, m-i)*Stirling_1(n+i+1,i+1), for 0 <= m <= n.

A156529 0 FC A triangular sequence: e(n,k)=(k + 1)e(n - 1, k) + (2n - k - 1)e(n - 1, k - 1); t(n,m)=(e(n, k)*e(n, n - k - 1)).

A156529 0 LD A triangular sequence: e(n,k)=(k + 1)e(n - 1, k) + (2n - k - 1)e(n - 1, k - 1); t(n,m)=(e(n, k)*e(n, n - k - 1)).

A156540 0 LD An anti-diagonal triangular sequence from the "blended" q-factorial: t(n,m)=If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].

A156564 0 LD A triangular sequence of the anti-diagonal of the Bonacci q like factorial: t(n,m)=If[m == 0, n!, Product[(m + 1)^n - Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].

A156565 0 LD A triangular sequence of the anti-diagonal of the Cyclotomic q like factorial: t(n,m)=If[m == 0, n!, Product[Cyclotomic[k, m + 1], {k, 1, n}]].

A156576 0 LD A triangular sequence of the anti-diagonal of the q squared like factorial: t(n,m)=If[m == 0, n!, Product[Sum[(i + 1)*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].

A156579 0 LD A triangular sequence of the anti-diagonal of the backward q squared like factorial: t(n,m)=If[m == 0, n!, Product[Sum[(k - i)*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].

A156582 0 LD Anti-diagonal triangle sequence of binomial q combinations: t(n,k)=If[m == 0, n!, Product[Sum[ Binomial[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].

A156588 0 LD A triangle of q factorial type based on Stirling first polynomials: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].

A156603 0 LD A q-factorial triangle sequence built of Cartan A_n polynomials as anti-diagonals: p(x,n)=CartanAn(x,n): t(n,k)=If[m == 0, n!, Product[p(m+1),k), {k, 1, n}]];

A156612 0 LD A q-factorial triangle sequence built of Cartan D_n polynomials as anti-diagonals: p(x,n)=CartanDn(x,n): t(n,k)=If[m == 0, n!, Product[p(m+1),k), {k, 1, n}]];

A156647 0 LD A q factorial based on Shabat ChebyshevT (*A123583*) Polynomials as anti-diagonals: t(n,k)=If[m == 0, n!, Product[1 - ChebyshevT[k, m + 1]^2, {k, 1, n}]].

A156693 0 LD General qp weighted factorial as antidiagonals: t(n,m)=If[m == 0, n!, Product[Product[1 - (i + 1)*(m + 1), {i, 0, k - 1}], {k, 1, n}]].

A156699 0 LD General qp odd weighted factorial as antidiagonals: t(n,m)=If[m == 0, n!, Product[Product[1 - (2*i - 1)*( m + 1), {i, 0, k - 1}], {k, 1, n}]].

A156730 0 LD General qp linear weighted factorials as : t(n,m)=If[m == 0, n!, Product[Product[1 - (3*i - 2)*( m + 1), {i, 0, k - 1}], {k, 1, n}]],

A156815 0 LD Triangle read by rows: t(n,m)=n!*StirlingS2[n, m]/Binomial[n, m].

A156815 0 SC Triangle read by rows: t(n,m)=n!*StirlingS2[n, m]/Binomial[n, m].

A156881 0 LD Fibonacci base q-factorials as anti-diagonals: t(n,m)=If[m == 0, n!, Product[Sum[(-( m + 1) + (m + 1)^2)^i, {i, 0, k - 1}], {k, 1, n}]].

A156882 0 LD Padovan/ Minimal Pisot base q-factorials as anti-diagonals: t(n,m)=If[m == 0, n!, Product[Sum[(-( m + 1) + (m + 1)^3)^i, {i, 0, k - 1}], {k, 1, n}]].

A156883 0 LD Higher Minimal Pisot base q-factorials as anti-diagonals: t(n,m)=If[m == 0, n!, Product[Sum[(-( m + 1)^3 + (m + 1)^4)^i, {i, 0, k - 1}], {k, 1, n}]].

A156885 0 LD Higher (A097886) Pisot base q-factorials as anti-diagonals: t(n,m)=If[m == 0, n!, Product[Sum[((m + 1)^2 - (m + 1)^3 - (m + 1)^4 + (m + 1)^5)^i, {i, 0, k - 1}], {k, 1, n}]].

A156888 0 LD Higher (A107479) Pisot base q-factorials as anti-diagonals: t(n,m)=If[m == 0, n!, Product[Sum[(-( 1 + m + (1 + m)^2 + (1 + m)^3 + (1 + m)^4 + (1 + m)^5 - (1 + m)^7))^i, {i, 0, k - 1}], {k, 1, n}]].

A156889 0 LD Higher Pisot base q-factorials as anti-diagonals: t(n,m)=If[m == 0, n!, Product[Sum[((1 + m)^2 - (1 + m)^4 - (1 + m)^5 + (1 + m)^6)^i, {i, 0, k - 1}], {k, 1, n}]].

A156953 0 LD Adjusted general Roman-Appell -Andrews q-factorials: t(n,k)=If[m == 0, n!, Product[Product[1 - (m + 1)^i, {i, 1, k}]/((1 - (m + 1))^k), {k, 1, n}]].

A156991 0 SC A triangular sequence:t(n,m)=n!*Binomial[n + m - 1, n]

A156992 0 FC Triangle T(n,k) = n!*binomial(n-1,k-1) read by rows, 1<=k<=n.

A156992 0 LD Triangle T(n,k) = n!*binomial(n-1,k-1) read by rows, 1<=k<=n.

A156995 0 FC A triangle sequence from polynomial coefficients: p(x,n)=If[n == 0, 1, Sum[Binomial[2*n - m, m]*(n - m)!*(2*n/(2*n - m))x^m, {m, 0, n}]].

A157047 0 LD A triangle of infinite sum coefficients with: Limit[Log[1-x],x->0]=-x: p(x,y)=1+n!*x^(n - 1)*Sum[x^k/(k*Binomial[n + k, k]), {k, 1, Infinity}]; such that Log[1-x]->-x.

A157400 0 LD A partition product with biggest-part statistic of Stirling_1 type (with parameter k = -2) as well as of Stirling_2 type (with parameter k = -2), (triangle read by rows).

A157400 0 SD A partition product with biggest-part statistic of Stirling_1 type (with parameter k = -2) as well as of Stirling_2 type (with parameter k = -2), (triangle read by rows).

A157743 0 SD A recursion triangle sequence based on the Eulerian numbers: A(n,k)=n*A(n-1,k-1)+k*Eulerian(n-1,k).

A158359 0 LD Triangle T(n,k) read by rows: coefficient [x^(n-k)] of the characteristic polynomial of the n X n matrix A(r,c)=1 (if c>r) and A(r,c)=c (if c<=r).

A158389 0 FC A triangle of matrix polynomials: M(d)=Reverse[Table[If[ m <= n, m, 0], {n, 1, d}, {m, 1, d}]].

A158471 0 LD Stirling-like triangle by rows generated from (x-1)*(x-1)*(x-2)*(x-3)*(x-4)*...

A158868 0 LD A polynomial integration sequence: t(n,m)=((2*n + 1)!!/(2^(Floor[(n - 1)/ 2] + Floor[m/2] + 1)))Integrate[(2 - 2*x)^(Floor[(n - 1)/2] + Floor[m/2] + 1)*(2*x)^(Floor[(m - 1)/2] + Floor[n/2] + 1), {x, 0, 1}].

A161119 0 LD Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k cycles with entries of opposite parities (0<=k<=n).

A161129 0 LD Triangle read by rows: T(n,k) is the number of non-derangements of {1,2,...,n} for which the difference between the largest and smallest fix points is k (n>=1; 0<=k<=n-1).

A162508 0 LD A binomial sum of powers related to the Bernoulli numbers, triangular array, read by rows.

A162608 0 FC Triangle read by rows in which row n lists n+1 terms, starting with n!, such that the difference between successive terms is also equal to n!.

A162608 0 LD Triangle read by rows in which row n lists n+1 terms, starting with n!, such that the difference between successive terms is also equal to n!.

A163626 0 LD Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y=1/(1+exp(-x))

A163936 0 FC Triangle related to the o.g.f.s. of the right hand columns of A130534 (E(x,m=1,n))

A163937 0 FC Triangle related to the o.g.f.s. of the right hand columns of A028421 (E(x,m=2,n))

A165490 0 SC Triangle read by rows, A084938 * A165489 diagonalized as an infinite lower triangular matrix.

A165675 0 FC Extended triangle related to the asymptotic expansions of the E(x,m=2,n)

A165680 0 LD Triangle of the divisors of the coefficients of triangles A138771 and A165675

A165680 0 SD Triangle of the divisors of the coefficients of triangles A138771 and A165675

A166350 0 LD Table T(n,m) = m! read by rows.

A166350 0 SD Table T(n,m) = m! read by rows.

A167546 0 FC The ED1 array read by antidiagonals

A167556 0 FC A triangle related to the GF(z) formulas of the rows of the ED1 array A167546.

A167556 0 SC A triangle related to the GF(z) formulas of the rows of the ED1 array A167546.

A167557 0 FC The lower left triangle of the ED1 array A167546.

A167560 0 FC The ED2 array read by antidiagonals

A167568 0 LD A triangle related to the GF(z) formulas of the rows of the ED2 array A167560.

A167569 0 FC The lower left triangle of the ED2 array A167560.

A168295 0 FC Worpitzky form polynomials for the {1,8,1} A142458 sequence: p(x,n) = Sum[A(n, k)*Binomial[x + k - 1, n - 1], {k, 1, n}]

A168296 0 FC Worpitzky form polynomials for the {1,16,1} A142462 sequence: p(x,n) = Sum[A(n, k)*Binomial[x + k - 1, n - 1], {k, 1, n}]

A168391 0 FC Worpitzky form polynomials for the Narayana triangle A001263(n,k):p(x,n) = Sum[A001263(n,k)*Binomial[x + k - 1, n - 1], {k, 1, n}]

A169593 0 FC Coefficients of characteristic polynomials of determinant equals trace matrices using Eulerian trace and factorial determinant.

A169593 0 SD Coefficients of characteristic polynomials of determinant equals trace matrices using Eulerian trace and factorial determinant.

A173333 0 FC Triangle read by rows: T(n,k) = n! / k!, 1<=k<=n.

A174298 0 FC A symmetrical triangle: t(n,m)=Binomial[n, m]*If[Floor[n/2] greater than or equal to m, n!/m!, n!/(n - m)! ]

A174298 0 LD A symmetrical triangle: t(n,m)=Binomial[n, m]*If[Floor[n/2] greater than or equal to m, n!/m!, n!/(n - m)! ]

A174551 0 LD Triangular array T(n,k): functions f:{1,2,...,n}-> {1,2,...,n} such that each of k fixed (but arbitrary) elements are in the image of f.

A174552 0 LD Triangular array T(n,k): The differences in the columns of A174551.

A174893 0 FC Coefficients of expansion polynomial:p(x,t)=Exp[ -t^2* x](1 - t)^(-x)/x

A176013 0 FC Triangle read by rows:t(n,m)=(-1)^n*(n!/m!)*(Binomial[n - 1, m - 1]*Binomial[n, m - 1]/m)

A176989 1 SD Triangle read by rows: the coefficient [t^n x^k] of n!*(n+2)! *exp(x*t) *(t*(1-2*Exp(t))-2*Exp(t)) / (2*(1-exp(t))), in row n, k=0..n+1.

A176990 1 SD Table T(n,m) read by rows: the coefficient of [t^n x^m] of 2*n!*(n+2)!*exp(x*t)*( t*(1-exp(t))-exp(t) ) / (1-exp(t) ), 0<=m<=n+1.

A177428 0 LD Triangle T(n,m)= A141686(n,m)*(m-1)! read by rows, n>=1, 1<=m<=n.

A177429 0 LD Triangle read by rows: T(n,m)=A060187(1+n,1+m) *n! / (n-m)!

A177847 0 FC Array T(n,m)= (n*m)!*Beta(n, m) read by antidiagonals.

A177847 0 LD Array T(n,m)= (n*m)!*Beta(n, m) read by antidiagonals.

A178126 0 SC A triangle of polynomial coefficients:p(x,n)=If[n == 0, 1, n!*(Binomial[x + n, n] - Binomial[x, n])]

A179455 0 FC Triangle read by rows: number of permutation trees of power n and height <= k + 1.

A179456 0 SC Triangle read by rows: number of permutation trees of power n and height <= n - k.

A179457 0 LD Triangle read by rows: number of permutation trees of power n and width <= k.

A180397 0 FC T(n,m) = binomial(m!,n).

A181320 0 FC Triangle T(n,m) read by rows: the number of series-parallel networks with n+2 vertices and m+n+1 edges.

A181511 0 LD Triangle T(n,k) = n!/(n-k)! read by rows, 0<=k<n.

A181731 0 SC Table A(d,n) of the number of paths of a chess rook in a d-dimensional hypercube from (0...0) to (n...n) where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).

A182062 0 FC T(n,k) = C(n+1-k,k)*k!*(n-k)!, the number of ways for k men and n-k women to form a queue in which no man is next to another man.

A182062 0 SC T(n,k) = C(n+1-k,k)*k!*(n-k)!, the number of ways for k men and n-k women to form a queue in which no man is next to another man.

A185421 0 LD Ordered forests of k increasing unordered trees on the vertex set {1,2,...,n} in which all outdegrees are <= 2.

A185423 0 LD Ordered forests of k increasing plane unary-binary trees with n nodes.

A185815 0 FC Exponential Riordan array (log(1/(1-x)),x*A005043(x)).

A185896 0 LD Triangle of coefficients of (1/sec^2(x))*D^n(sec^2(x)) in powers of t = tan(x), where D = d/dx.

A187555 0 SC Triangle read by rows, defined by T(n,k)=binomial(n,k)*|Stirling1(n,k)|, 0<=k<=n.

A187556 0 SC Triangle read by rows of products of (signless) Stirling numbers of the first kind (A132393) and Stirling numbers of the second kind (A008277).

A187783 0 SD De Bruijn's triangle, T(m,n) = (m*n)!/(n!^m) read by antidiagonals.

A187784 0 LD Triangular array read by rows: T(n,k) is the number of ordered set partitions of {1,2,...,n} with exactly k singletons, n>=0, 0<=k<=n.

A188808 0 FC T(n,k)=Number of nXk array permutations with each element remaining in its original row or its original column

A188808 0 LD T(n,k)=Number of nXk array permutations with each element remaining in its original row or its original column

A188881 0 FC Triangle of coefficients arising from an expansion of Integral( exp(exp(exp(x))), dx).

A188881 0 LD Triangle of coefficients arising from an expansion of Integral( exp(exp(exp(x))), dx).

A190782 0 FC Triangle T(n,k), read by rows, of the coefficients of x^k in the expansion of sum_(m= 0..n) binomial(x,m)) = (a(k)*x^k)/n!, n >= 0, 0 <= k <= n.

A193093 0 FC Augmentation of the triangular array P=A094727 given by p(n,k)=n+k+1 for 0<=k<=n. See Comments.

A193094 0 FC Augmentation of the triangular array P=A130296 whose n-th row is (n+1,1,1,1,1...,1) for 0<=k<=n. See Comments.

A193229 0 LD A double factorial triangle

A193344 0 LD Triangle read by rows: T(n,m) (n>=0, 1 <= m <= n+1) = number of unlabeled rigid interval posets with n non-maximal and m maximal elements.

A193474 0 FC Table read by rows: The coefficients of the polynomials P(n,x) = Sum{k=0..n}Sum{j=0..k}(-1)^j*2^(-k)*C(k,j)*(k-2*j)^n*x^(n-k).

A193561 0 FC Augmentation of the triangle A004736. See Comments.

A196347 0 FC Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

A196347 0 LD Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

A196776 0 LD Triangle T(n,k) counts the number of ordered partitions of an n set into k odd-sized blocks.

A199220 0 LD Triangle read by rows: T(n,k)=(n-1-k)*|s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind and 1<=k<=n.

A199221 0 LD Triangle read by rows: T(n,k)=(n-1-k)*|s(n,n+1-k)|-2*|s(n-1,n-k)|, where s(n,k) are the signed Stirling numbers of the first kind and 1<=k<=n.

A199400 0 LD Triangle T(n,k), read by rows, given by (2,0,3,0,4,0,5,0,6,0,7,0,8,0,9,...) DELTA (2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,...) where DELTA is the operator defined in A084938.

A199673 0 LD Ways to form k labelled groups, each with a distinct leader, using n people. Triangle T(n,k)=n!*k^(n-k)/(n-k)! for 1<=k<=n.

A200265 0 FC Triangle read by rows: coefficients in an asymptotic expansion of the n-th prime.

A201637 0 SD Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.

A201685 0 LD Triangular array read by rows. T(n,k) is the number of connected endofunctions on {1,2,...,n} that have exactly k nodes in the unique cycle of its digraph representation.

A201685 0 SD Triangular array read by rows. T(n,k) is the number of connected endofunctions on {1,2,...,n} that have exactly k nodes in the unique cycle of its digraph representation.

A202363 0 SD Triangular array read by rows: T(n,k) is the number of inversion pairs ( p(i) < p(j) with i>j ) that are separated by exactly k elements in all n-permutations (where the permutation is represented in one line notation); n>=2, 0<=k<=n-2.

A204024 0 LD Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(i(i+1)/2, j(j+1)/2) (A106255).

A204126 0 LD Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).

A206703 0 LD Triangular array read by rows. T(n,k) is the number of partial permutations (injective partial functions) of {1,2,...,n} that have exactly k elements in a cycle. The k elements are not necessarily in the same cycle. A fixed point is considered to be in a cycle.

A206703 0 SD Triangular array read by rows. T(n,k) is the number of partial permutations (injective partial functions) of {1,2,...,n} that have exactly k elements in a cycle. The k elements are not necessarily in the same cycle. A fixed point is considered to be in a cycle.

A206823 0 LD Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} with exactly k elements x such that |f^(-1)(x)| = 1; n>=0, 0<=k<=n.

A208058 0 FC Triangle by rows relating to the factorials, generated from A002260.

A208183 0 SC Number of distinct k-colored necklaces with n beads per color; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A208447 1 SD Sum of the k-th powers of the numbers of standard Young tableaux over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A211365 0 FC Array read by antidiagonals: T(m,n) = Sum(1<=i<=m) i * (2m+n-1-i)!

A211366 0 FC Array read by antidiagonals: T(m,n) = Sum(1<=i<=m) i * ( n + 2(i-1) )!

A211367 0 FC Array read by antidiagonals: T(m,n) = m * Sum(1<=i<=m) (m+n-2+i)!

A211368 0 FC Array read by antidiagonals: T(m,n) = Sum(1<=i<=m) ( n + 2(i-1) )!

A211369 0 FC Array read by antidiagonals: T(m,n) = m*(m+n-1)! + Sum( n <= i <= m+n-2 ) i!

A211370 0 FC Array read by antidiagonals: T(m,n) = Sum( n <= i <= m+n-1 ) i!.

A211603 0 FC Triangular array read by rows: T(n,k) is the number of n-permutations that are pure cycles having exactly k fixed points; n>=2, 0<=k<=n-2.

A213275 0 SC Number A(n,k) of words w where each letter of the k-ary alphabet occurs n times and for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A213279 0 FC Triangle read by rows: T(n,k) (n>=1, 1 <= k <= n) = number of permutations of [1..n] in which all cycle lengths are divisible by k.

A213279 0 LD Triangle read by rows: T(n,k) (n>=1, 1 <= k <= n) = number of permutations of [1..n] in which all cycle lengths are divisible by k.

A213934 0 LD Triangle with entry a(n,m) giving the number of necklaces of n beads (C_N symmetry) with n colors available for each bead, but only m distinct fixed colors, say c[1],...,c[m], are present, with m from {1,...,n} and n>=1.

A213935 0 LD Triangle with entry a(n,m) giving the total number of necklaces of n beads (C_n symmetry) with n colors available for each bead, but only m distinct colors present, with m from {1, 2, ..., n} and n >= 1.

A213936 0 FC Number triangle with entry a(n,k), n>=1, m=1, 2, ..., n, giving the number of representative necklaces with n beads (C_n symmetry) corresponding to the color multinomial c[1]^k*c[2]*...*c[n+1-k].

A214152 0 FC Number of permutations T(n,k) in S_n containing an increasing subsequence of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

A214178 0 SD Triangle T(n,k) by rows: the k-th derivative of the Fibonacci Polynomial F_n(x) evaluated at x=1.

A214435 0 LD Triangle read by rows: T(n,k) = n!*S(n,k), where S(n,k) is the matrix inverse of the triangle zeta(k-n,1) - zeta(k-n,k+1), n>=1, k>=1.

A215216 0 FC Coefficient triangle of the Hermite-Bell polynomials for power -2.

A216242 0 FC Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} with a height of k; n>=1, 0<=k<=n-1.

A216242 0 LD Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} with a height of k; n>=1, 0<=k<=n-1.

A216919 0 FC The Gauss factorial N_n! for N >= 0, n >= 1, square array read by antidiagonals.

A216971 0 FC Triangle read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} that have exactly k nonrecurrent elements mapped to some (one or more) recurrent element. n >= 1, 0 <= k <= n-1.

A217629 0 LD Triangle, read by rows, where T(n,k) = k!*C(n, k)*3^(n-k) for n>=0, k=0..n.

A217877 0 FC Triangle read by rows: minimum inversion terminator in rooted labeled trees.

A217877 0 LD Triangle read by rows: minimum inversion terminator in rooted labeled trees.

A217891 0 LD T(n,k)=Number of n element 1..n arrays with each element the minimum of k adjacent elements of a permutation of 1..n+k-1 of n+k-1 elements

A217922 0 SC Triangle read by rows: labeled trees counted by improper edges.

A218016 0 LD Triangle, read by rows, where T(n,k) = k!*C(n, k)*5^(n-k) for n>=0, k=0..n.

A218017 0 LD Triangle, read by rows, where T(n,k) = k!*C(n, k)*7^(n-k) for n>=0, k=0..n.

A218018 0 LD Triangle, read by rows, where T(n,k) = k!*C(n, k)*11^(n-k) for n>=0, k=0..n.

A219374 0 LD Triangle of F(n,r) of r-geometric numbers, 1<=r<=n.

A219570 0 FC Triangular array read by rows. T(n,k) is the number of necklaces (turning over is not allowed) of n labeled black or white beads having exactly k black beads.

A219570 0 LD Triangular array read by rows. T(n,k) is the number of necklaces (turning over is not allowed) of n labeled black or white beads having exactly k black beads.

A219570 0 SC Triangular array read by rows. T(n,k) is the number of necklaces (turning over is not allowed) of n labeled black or white beads having exactly k black beads.

A219570 0 SD Triangular array read by rows. T(n,k) is the number of necklaces (turning over is not allowed) of n labeled black or white beads having exactly k black beads.

A219694 0 FC Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k nonrecurrent elements; n>=1, 0<=k<=n-1.

A219859 0 FC Triangular array read by rows: T(n,k) is the number of endofunctions, functions f:{1,2,...,n}->{1,2,...,n}, that have exactly k elements with no preimage; n>=0, 0<=k<=n.

A220222 0 LD Triangular array read by rows. T(n,k) is the number of functional digraphs on {1,2,...,n} such that no node is at a distance greater than one from a cycle and there are k recurrent elements whose preimage contains only one element, n>=0, 0<=k<=n.

A220234 0 LD Triangular array read by rows. T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} that have exactly k recurrent elements whose preimage contains only one element, n>=0, 0<=k<=n.

A220883 0 FC Triangle read by rows: row n gives coefficients of expansion of Product_{k = 1..n-1} ((n + 1)*x + k), starting with lowest power.

A220884 0 FC Triangle read by rows: row n gives coefficients of expansion of Product_{k=2..n} ((n+1-k)*x+k), starting with lowest power.

A220884 0 SC Triangle read by rows: row n gives coefficients of expansion of Product_{k=2..n} ((n+1-k)*x+k), starting with lowest power.

A221438 0 FC T(n,k)=Number of nXk 1..(max n,k) arrays with each row and column having unrepeated values

A221438 0 LD T(n,k)=Number of nXk 1..(max n,k) arrays with each row and column having unrepeated values

A221623 0 FC T(n,k)=Number of nXk arrays with each row a permutation of 1..k having at least as many downsteps as the preceding row

A222005 0 FC T(n,k)=Number of nXk arrays with each row a permutation of 1..k having at least as many downsteps as the preceding row, with rows in lexicographically nonincreasing order

A222159 0 FC T(n,k)=Number of nXk arrays with each row a permutation of 1..k having at least as many downsteps as the preceding row, with rows in lexicographically nondecreasing order

A222864 0 LD Triangle T(n,k) of strongly graded (3+1)-free partially ordered sets (posets) on n labeled vertices with height k.

A222866 0 LD Triangle T(n,k) of weakly graded (3+1)-free partially ordered sets (posets) on n labeled vertices with height k.

A225094 0 SC Number A(n,k) of lattice paths without interior points from {n}^k to {0}^k using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A225213 0 LD Triangular array read by rows. T(n,k) is the number of cycles in the digraph representation of all functions f:{1,2,...,n}->{1,2,...,n} that have length k; 1<=k<=n.

A225475 0 LD Triangle read by rows, k!*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

A225476 0 LD Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

A225479 0 LD Triangle read by rows, the ordered Stirling cycle numbers, T(n, k) = k!* s(n, k); n >= 0 k >= 0.

A225479 0 SC Triangle read by rows, the ordered Stirling cycle numbers, T(n, k) = k!* s(n, k); n >= 0 k >= 0.

A225816 0 SC Square array A(n,k) = (k!)^n, n>=0, k>=0, read by antidiagonals.

A226874 0 LD Number T(n,k) of n-length words w over a k-ary alphabet {a1,a2,...,ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 1, where #(w,x) counts the number of letters x in word w; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A227550 0 FC A triangle formed like Pascal's triangle, but with factorial(n) on the borders instead of 1.

A227550 0 LD A triangle formed like Pascal's triangle, but with factorial(n) on the borders instead of 1.

A227655 0 SC Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_k) we have abs(p_{i}-p_{i+1}) <= 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A229565 0 LD T(n,k)=Number of arrays of length n that are sums of k consecutive elements of length n+k-1 permutations of 0..n+k-2

A231536 0 FC Triangular array read by rows. T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} whose functional digraph has exactly k nodes such that no nonrecurrent element is mapped into it. n>=1, 1<=k<=n.

A231536 0 LD Triangular array read by rows. T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} whose functional digraph has exactly k nodes such that no nonrecurrent element is mapped into it. n>=1, 1<=k<=n.

A231602 0 LD Triangular array read by rows: T(n,k) is the number of rooted labeled trees on n nodes that have exactly k nodes with outdegree = 1, n>=1, 0<=k<=n-1.

A233357 0 LD Triangle read by rows: T(n,k) = ((Stirling2)^2)(n,k) * k!

A233543 0 LD Table T(n,m) = m! read by rows.

A233543 0 SD Table T(n,m) = m! read by rows.

A235595 0 LD Triangle read by rows: the triangle in A034855, with the n-th row normalized by dividing it by n.

A238363 0 FC Coefficients for the commutator for the logarithm of the derivative operator [log(D),x^n D^n]=d[(xD)!/(xD-n)!]/d(xD) expanded in the operators :xD:^k.

A238385 0 FC Shifted lower triangular matrix A238363 with a main diagonal of ones.

A239098 0 SC Triangle read by rows: T(0,0)=1; T(m,0)=0; otherwise T(m,n) = (m-1)*T(m-1,n)+(m-1+n)*T(m-1,n-1).

A241981 0 LD Number T(n,k) of endofunctions on [n] where the largest cycle length equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A243098 0 LD Number T(n,k) of endofunctions on [n] with all cycles of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A243202 0 LD Coefficients of a particular decomposition of N^N in terms of binomial coefficients.

A243203 0 LD Terms of a particular integer decomposition of N^N.

A245334 0 LD A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

A245405 0 LD Number A(n,k) of endofunctions on [n] such that no element has a preimage of cardinality k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A246049 0 LD Number T(n,k) of endofunctions on [n] where the smallest cycle length equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A246072 0 SD Number A(n,k) of permutations p on [2n] satisfying p^k(i) = i for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A246609 0 LD Number T(n,k) of endofunctions on [n] whose cycle lengths are multiples of k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A247005 0 SD Number A(n,k) of permutations on [n] that are the k-th power of a permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A247497 0 LD Triangle read by rows, T(n,k) (n>=0, 0<=k<=n) coefficients of the partial fraction decomposition of rational functions generating the columns of A247495 (the Motzkin polynomials evaluated at nonnegative integers).

A247500 0 FC Triangle read by rows: T(n,k)=((k+1)*(n+1)*Gamma(n+1)^2)/(Gamma(k+2)^2 *Gamma(n-k+2)), 0<=k<=n.

A247501 0 LD Triangle read by rows, T(n,k) (n>=0, 0<=k<=n) coefficients of the partial fraction decomposition of rational functions generating the columns of A247498 (the Swiss-Knife polynomials evaluated at nonnegative integers).

A247504 0 SD Square array read by antidiagonals, A(n,k) = k!*[x^k]((1-sum_{j=1..n} x^j)^(-1)), (n>=0,k>=0).

A248090 0 LD Square array read by antidiagonals: T(n,k) is the number of k-edge colored trees on vertex set [n] (n>=2, k>=2).

A248686 0 LD Triangular array of multinomial coefficients : T(n,k) = n!/(n(1)!*n(2)!* ...*n(k)!), where n(i) = floor((n + i - 1)/k) for i = 1 .. k.

A249026 0 SD Array read by antidiagonals upwards: T(d,n) = number of d-dimensional permutations of n letters (d >= 0, n >= 1).

A249027 0 LD Array read by antidiagonals upwards: T(d,n) = number of d-dimensional permutations of n letters (d >= 1, n >= 1).

A249631 0 LD Number of permutations p of {1,...,n} such that |p(i+1)-p(i)| < k, k=2,...,n; T(n,k), read by rows.

A249796 0 FC Triangle T(n,k), n>=3, 3<=k<=n, read by rows. Number of ways to make n selections without replacement from a circular array of n unlabeled cells (ignoring rotations and reflection), such that the first selection of a cell adjacent to previously selected cells occurs on the k-th selection.

A253284 0 FC Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.

A253668 0 LD Square array read by ascending antidiagonals, T(n, k) = k!*[x^k](log(x+1)*sum(j=0..n, C(n,j)*x^j)), n>=0, k>=0.

A253669 0 LD Square array read by ascending antidiagonals, T(n, k) = k!*[x^k](log(x+1)*sum(j=0..n, C(2*n,j)*x^j)), n>=0, k>=0.

A254040 0 LD Number T(n,k) of primitive (=aperiodic) n-bead necklaces with colored beads of exactly k different colors; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A254876 0 LD Triangular table T(n,k) = n! / Product_{m=(n-floor((2n)/(3^k))) .. (n-floor((n)/(3^k)))} m, read by rows T(1,1), T(2,1), T(2,2), T(3,1), T(3,2), T(3,3), ...

A254876 0 SD Triangular table T(n,k) = n! / Product_{m=(n-floor((2n)/(3^k))) .. (n-floor((n)/(3^k)))} m, read by rows T(1,1), T(2,1), T(2,2), T(3,1), T(3,2), T(3,3), ...

A254883 0 SC Triangle read by rows, T(n,k) = sum(j=0..2*(n-k), A254882(n-k,j)*k^j /(n-k)!), n>=0, 0<=k<=n.

A254933 0 LD Triangle used for the integral of even powers of the sine and cosine functions.

A255008 0 SC Array T(n,k) read by ascending antidiagonals, where T(n,k) is the numerator of polygamma(n, 1) - polygamma(n, k).

A255970 0 LD Number T(n,k) of partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A256215 0 FC Triangle read by rows: T(n,k) = (n-1)!*n^(k-1)*binomial(n,k)/(k-1)!.

A256268 0 SC Table of k-fold factorials, read by antidiagonals.

A256548 0 SC Triangle read by rows, T(n,k) = |n,k|*h(k), where |n,k| are the Stirling cycle numbers and h(k) = hypergeom([-k+1,-k],[],1), for n>=0 and 0<=k<=n.

A257493 0 SD Number A(n,k) of n X n nonnegative integer matrices with all row and column sums equal to k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A257783 0 LD Number T(n,k) of words w of length n such that each letter of the k-ary alphabet is used at least once and for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A259334 0 LD Triangle read by rows: T(n,k) = k*(n-1)!*n^(n-k-1)/(n-k)!, 1 <= k <= n.

A259456 0 FC Triangle read by rows, giving coefficients in an expansion of absolute values of Stirling numbers of the first kind in terms of binomial coefficients.

A260322 0 LD Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=1.

A260322 0 SD Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=1.

A260323 0 LD Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=-1.

A260323 0 SD Triangle read by rows: T(n,k) = logarithmic polynomial G_k^(n)(x) evaluated at x=-1.

A260324 0 LD Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=1.

A260324 0 SD Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=1.

A260325 0 LD Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=-1.

A260325 0 SD Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=-1.

A260338 0 LD Triangle read by rows: Cayley's numbers phi(m,n) (m,n>=0). Row m contains phi(m,0), phi(m-1,1), phi(m-2,2), ..., phi(0,m).

A260338 0 SD Triangle read by rows: Cayley's numbers phi(m,n) (m,n>=0). Row m contains phi(m,0), phi(m-1,1), phi(m-2,2), ..., phi(0,m).

A260687 0 FC Triangular array with n-th row giving coefficients of polynomial Product_{k = 2..n} (k + n*t) for n >= 1.

A261902 1 SD Irregular triangle read by rows: T(n,m) = number of permutations of {1, 2, ..., n} which form arithmetic progressions modulo m (n>=1, 1<=m<=n+2).

A261964 0 FC Chocolate numbers read as a triangle across rows: T(n,k), n >= 1, 1 <= k <= n.

A261964 0 LD Chocolate numbers read as a triangle across rows: T(n,k), n >= 1, 1 <= k <= n.

A262071 0 SC Number T(n,k) of ordered partitions of an n-set with nondecreasing block sizes and maximal block size equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A264954 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no equal horizontal or vertical neighbors.

A265607 0 LD Triangle read by rows, T(n,k) = n!*B(n,k) for n>=0 and 0<=k<=n, where B(n,k) is the Bell matrix with generator 1/j for j>=1.

A265607 0 SC Triangle read by rows, T(n,k) = n!*B(n,k) for n>=0 and 0<=k<=n, where B(n,k) is the Bell matrix with generator 1/j for j>=1.

A265609 0 SD Array read by ascending antidiagonals: A(n,k) the rising factorial, also known as Pochhammer symbol, for n>=0 and k>=0.

A265856 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element plus any horizontal or antidiagonal neighbor equal to n-1.

A265870 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element plus any antidiagonal neighbor equal to n-1.

A265882 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element plus any horizontal neighbor equal to n-1.

A266183 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element plus any diagonal or antidiagonal neighbor equal to n-1.

A266309 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element 1 greater than its west neighbor modulo n and the upper left element equal to 0.

A266572 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element 1 greater than its northwest or southwest neighbor modulo n and the upper left element equal to 0.

A266655 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element 1 greater than its northeast neighbor modulo n and the upper left element equal to 0.

A266681 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element 1 greater than its northwest or northeast neighbor modulo n and the upper left element equal to 0.

A266867 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element 1 greater than its west or northeast neighbor modulo n and the upper left element equal to 0.

A266994 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element 1 greater than its west, northeast or northwest neighbor modulo n and the upper left element equal to 0.

A267019 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with every element equal to or 1 greater than any northeast or northwest neighbors modulo n and the upper left element equal to 0.

A267072 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element 1 greater than its west or southwest neighbor modulo n and the upper left element equal to 0.

A267163 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element 1 greater than its west, southwest or northwest neighbor modulo n and the upper left element equal to 0.

A267222 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with no element 1 greater than its west, northeast or southeast neighbor modulo n and the upper left element equal to 0.

A267278 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with every element equal to or 1 greater than any southwest or northwest neighbors modulo n and the upper left element equal to 0.

A267328 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with every element equal to or 1 greater than any southwest or northwest neighbors modulo n and the upper left element equal to 0.

A267425 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with every element equal to or 1 greater than any west, northeast or northwest neighbors modulo n and the upper left element equal to 0.

A267562 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with every element equal to or 1 greater than any west or southwest neighbors modulo n and the upper left element equal to 0.

A267617 0 SD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with every element equal to at least one horizontal or antidiagonal neighbor and the top left element equal to 0.

A267629 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with every element equal to or 1 greater than any west neighbor modulo n and the upper left element equal to 0.

A267724 0 SD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with every element equal to at least one horizontal neighbor and the top left element equal to 0.

A267742 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with every element equal to or 1 greater than any west or northeast neighbors modulo n and the upper left element equal to 0.

A267836 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with every element equal to or 1 greater than any northeast neighbor modulo n and the upper left element equal to 0.

A268216 0 LD Triangle read by rows: T(n,k) (n>=2, k=3..n+1) = number of partial chain topologies with respect to a single point on a set of n points and having k open sets.

A268472 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with column sums equal.

A269336 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 avoiding the pattern equal-up in every row and column.

A269565 0 FC T(n,m) = Number of (directed) Hamiltonian paths in K_{n} X K_{m}.

A269565 0 LD T(n,m) = Number of (directed) Hamiltonian paths in K_{n} X K_{m}.

A269646 0 LD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 avoiding the pattern down-up in every row and equal-up in every column.

A269722 0 FC Triangle read by rows: traces of web-coloring matrices.

A269722 0 LD Triangle read by rows: traces of web-coloring matrices.

A269940 0 SC Triangle read by rows, T(n,k) = Sum_{m=0..k} (-1)^(m+k)*C(n+k,n+m)*Stirling1(n+m,m), for n>=0, 0<=k<=n.

A271703 0 SC Triangle read by rows, the unsigned Lah numbers T(n,k) = binomial(n-1,k-1)*n!/k! if n>0 and k>0, T(n,0) = 0^n and otherwise 0, for n>=0 and 0<=k<=n.

A271707 0 LD Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with length k and Aut(p) = 1^j[1]*j[1]!*...*n^j[n]*j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.

A271708 0 SC Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with largest part k and Aut(p) = 1^j[1]*j[1]!*...*n^j[n]*j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.

A273730 0 LD Square array read by antidiagonals: A(n,k) = number of permutations of n elements divided by the number of k-ary heaps on n+1 elements, n>=0, k>=1.

A275062 0 SD Number A(n,k) of permutations p of [n] such that p(i)-i is a multiple of k for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.


different prefix

A145879 0 FC Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having exactly k entries that are midpoints of 321 patterns (0<=k<=n-2 for n>=2; k=0 for n=1). {1, 2, 1, 2, 6, 24, 120, 720, 5040}


A202992 1 SD Triangle T(n,k), read by rows, given by (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (0, 0, 1, 1, 2, 2, 3, 3, 4, 4, ...) where DELTA is the operator defined in A084938. {1, 2, 1, 2, 6, 24, 120, 720, 5040}

A211871 0 LD Number T(n,k) of permutations of n elements with no fixed points and largest cycle of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. {1, 0, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880}


A225540 0 FC Triangle of functions in a size n set for which the sequence of composition powers starts with a length k stem (index) before entering a cycle (period). {1, 4, 6, 24, 120, 720, 5040}

A243098 0 SD Number T(n,k) of endofunctions on [n] with all cycles of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. {0, 3, 6, 24, 120, 720, 5040, 40320}

A246049 0 SD Number T(n,k) of endofunctions on [n] where the smallest cycle length equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. {0, 3, 6, 24, 120, 720, 5040, 40320}

A246609 0 SD Number T(n,k) of endofunctions on [n] whose cycle lengths are multiples of k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. {0, 4, 6, 24, 120, 720, 5040, 40320}

A253668 0 SD Square array read by ascending antidiagonals, T(n, k) = k!*[x^k](log(x+1)*sum(j=0..n, C(n,j)*x^j)), n>=0, k>=0. {0, 1, 1, 1, 2, 6, 24, 120, 720, 5040}


possible typos

near misses, variations and coincidences

Factorial numbers:

1,1,2,6,24,120,720,5040,40320,362880,3628800,39916800,479001600,6227020800,87178291200,1307674368000, ...


A100711 0 FC Table read by antidiagonals: T(m,n) gives the ordinal number in the table of permutations of length n+1 of the permutation which reverses the first m+1 items on a list of length n+1, leaving the remaining items unaltered. For example, T(5,7) is 28494 and the 28494th row of the permutation table of order 8 is 5 4 3 2 1 0 6 7. {1, 2, 6, 24, 120, 720, 5040, 40320, 85680, 4166}

A130561 1 SD Numbers associated to partitions, used for combinatoric interpretation of Lah triangle numbers A105278. {1, 6, 24, 120, 720, 120, 2520, 42}

A143474 0 FC a(n) = smallest k such that A141501(k) = 2*n+1. {1, 2, 6, 24, 120, 720, 3360, 15120}


A173869 1 SD Irregular table T(n,k) = A164341(n,k) * A036039(n,k) read by rows. {1, 6, 24, 120, 720, 480, 5040, 1092}

A178801 0 LD Write n! p(n) times. {1, 2, 6, 24, 120, 720, 720, 5040, 40320}

A178801 0 SD Write n! p(n) times. {2, 6, 24, 120, 720, 720, 5040, 5040}


A214078 0 LD a(n) = (ceiling (sqrt(n)))!. {1, 2, 6, 6, 24, 120, 720, 720, 5040}

A214078 0 SD a(n) = (ceiling (sqrt(n)))!. {1, 2, 6, 24, 120, 720, 720, 5040}

A214080 0 FC a(n) = (floor(sqrt(n)))! {1, 1, 1, 2, 6, 6, 24, 120, 720, 720}


A226441 3 SD T(n,k)=Number of permutations of 1..n with fewer than k interior elements having values lying between the values of their neighbors {0, 1, 4, 1, 2, 6, 24, 120, 720, 5038}


A248842 3 SD T(n,k)=Number of length n arrays x(i), i=1..n with x(i) in 0..i and no value appearing more than k times {0, 2, 8, 2, 6, 24, 120, 720, 5038}


A254876 0 SC Triangular table T(n,k) = n! / Product_{m=(n-floor((2n)/(3^k))) .. (n-floor((n)/(3^k)))} m, read by rows T(1,1), T(2,1), T(2,2), T(3,1), T(3,2), T(3,3), ... {1, 2, 6, 6, 24, 120, 720, 6480, 50400}

A263752 3 SD T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and every three consecutive elements having its maximum within 5 of its minimum {0, 1, 3, 1, 2, 6, 24, 120, 720, 1560}

A265089 0 SD T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with row sums equal and column sums equal. {1, 2, 6, 24, 120, 720, 6300, 45360, 589680}