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Triangles Diagonal CatalanNumbers

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Regular linear triangles having the catalan numbers as a diagonal or a column

The number following the Anum is the offset added to the sequence before putting it into a linear triangle.

FC: first column

SC: second column

LD: last diagonal

SD: second to last diagonal


A008550 0 SC Table T(n,k), n>=0 and k>=0, read by antidiagonals: the k-th column given by the k-th Narayana polynomial.

A008828 0 LD Triangle read by rows: T(n,k) = number of closed meander systems of order n with k<=n components.

A009766 0 LD Catalan's triangle T(n,k) (read by rows): each term is the sum of the entries above and to the left, i.e., T(n,k) = Sum_{j=0..k} T(n-1,j).

A009766 0 SD Catalan's triangle T(n,k) (read by rows): each term is the sum of the entries above and to the left, i.e., T(n,k) = Sum_{j=0..k} T(n-1,j).

A028364 0 FC Triangle T(n,m) = Sum_{k=0..m} Catalan(n-k)*Catalan(k).

A028364 0 LD Triangle T(n,m) = Sum_{k=0..m} Catalan(n-k)*Catalan(k).

A028376 0 LD Triangle read by rows: T(n,m) = Sum Catalan(n-k)*Catalan(k), k=0..m.

A028376 0 SC Triangle read by rows: T(n,m) = Sum Catalan(n-k)*Catalan(k), k=0..m.

A028378 0 SC Concatenate rows of triangle in A028364 (removing duplicates).

A030237 0 LD Catalan's triangle with right border removed.

A033184 0 FC Catalan triangle A009766 transposed.

A033184 0 SC Catalan triangle A009766 transposed.

A033282 0 LD Triangle read by rows: T(n,k) is the number of diagonal dissections of a convex n-gon into k+1 regions.

A033820 0 FC Triangle read by rows: T(k,j) = ((2*j+1)/(k+1))*binomial(2*j,j)*binomial(2*k-2*j,k-j).

A039598 0 FC Triangle formed from odd-numbered columns of triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x). Sometimes called Catalan's triangle.

A039599 0 FC Triangle formed from even-numbered columns of triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

A046527 0 FC A triangle related to A000108 (Catalan) and A000302 (powers of 4).

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

A047888 0 SC Rectangular array of numbers a(n,k) = number of permutations of n things with longest increasing subsequence of length <=k (1<=k<=infinity), read by antidiagonals.

A050144 0 FC T(n,k)=M(2n-1,n-1,k-1), 0<=k<=n, n >= 0, where M(p,q,r)=number of upright paths from (0,0) to (p,p-q) that meet the line y=x-r and do not rise above it.

A050145 0 FC T(n,k)=M(2n,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

A050157 0 FC T(n,k)=S(2n,n,k), 0<=k<=n, n >= 0, where S(p,q,r)=number of upright paths from (0,0) to (p,p-q) that do not rise above the line y=x-r.

A050157 0 SC T(n,k)=S(2n,n,k), 0<=k<=n, n >= 0, where S(p,q,r)=number of upright paths from (0,0) to (p,p-q) that do not rise above the line y=x-r.

A050158 0 FC T(n,k)=S(2n+1,n,k+1), 0<=k<=n, n >= 0, array S as in A050157.

A050159 0 FC T(n,k)=S(2n-1,n-1,k-1), 0<=k<=n, n >= 0, array S as in A050157.

A050159 0 SC T(n,k)=S(2n-1,n-1,k-1), 0<=k<=n, n >= 0, array S as in A050157.

A050160 0 FC T(n,k)=S(2n,n-1,k-1), 0<=k<=n, n >= 0, array S as in A050157.

A050165 0 LD Triangle read by rows: T(n,k)=M(2n+1,k,-1), 0<=k<=n, n >= 0, array M as in A050144.

A050166 0 LD Triangle T(n,k)=M(2n,k,-1), 0<=k<=n, n >= 0, array M as in A050144.

A053979 0 LD Triangle T(n,k) giving number of rooted maps regardless of genus with n edges and k nodes (n >= 0, k=1..n+1).

A054445 0 FC Triangle read by rows giving partial row sums of triangle A033184(n,m), n >= m >= 1 (Catalan triangle).

A059346 0 LD Difference array of Catalan numbers A000108 read by antidiagonals.

A059365 0 SC Another version of the Catalan triangle: T(r,s) = binomial(2*r-s-1,r-1)-binomial(2*r-s-1,r), r>=0, 0<=s<=r.

A060693 0 FC Triangle T(n, k) (0 <= k <= n) read by rows; given by [1, 1, 1, 1, 1, ...] DELTA [1, 0, 1, 0, 1, 0, ....] where DELTA is the operator defined in A084938.

A060854 0 SC Array T(m,n) read by antidiagonals: T(m,n) (m >= 1, n >= 1) = number of ways to arrange the numbers 1,2,..,m*n in an m X n matrix so that each row and each column is increasing.

A060854 0 SD Array T(m,n) read by antidiagonals: T(m,n) (m >= 1, n >= 1) = number of ways to arrange the numbers 1,2,..,m*n in an m X n matrix so that each row and each column is increasing.

A062991 0 FC Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x).

A062991 0 LD Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x).

A062993 0 FC A triangle (lower triangular matrix) composed of Pfaff-Fuss (or Raney) sequences.

A064045 0 SC Square array read by antidiagonals of number of length 2k walks on an n-dimensional hypercubic lattice starting and finishing at the origin and staying in the nonnegative part.

A064094 0 SC Triangle composed of generalized Catalan numbers.

A064308 0 LD Product of two triangular matrices C*S2.

A064879 0 SC Triangle of numbers composed of certain generalized Catalan numbers.

A067310 0 LD Square table read by antidiagonals of number of ways of arranging n chords on a circle with k simple intersections (i.e. no intersections with 3 or more chords).

A067323 0 FC Catalan triangle A028364 with row reversion.

A067323 0 LD Catalan triangle A028364 with row reversion.

A067345 0 FC Square array read by antidiagonals: T(n,k)=(T(n,k-1)*n^2-Catalan(k-1))/(n-1) with a(n,1)=1 and a(1,k)=Catalan(k) where Catalan(k)=C(2k,k)/(k+1)=A000108(k).

A067347 0 SC Square array read by antidiagonals: T(n,k)=(T(n,k-1)*n^2-Catalan(k-1)*n)/(n-1) with a(n,0)=1 and a(1,k)=Catalan(k) where Catalan(k)=C(2k,k)/(k+1)=A000108(k).

A070914 0 SD Array read by antidiagonals giving number of paths up and left from (0,0) to (n,kn) where x/y<=k for all intermediate points.

A076037 0 LD Square array read by antidiagonals in which row n has g.f. (1-(n-1)*x*C)/(1-n*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

A076037 0 SD Square array read by antidiagonals in which row n has g.f. (1-(n-1)*x*C)/(1-n*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

A076038 0 LD Square array read by antidiagonals in which row n has g.f. C/(1-n*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

A076038 0 SD Square array read by antidiagonals in which row n has g.f. C/(1-n*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

A078391 0 FC Triangle read by rows: T(n,k) = Catalan(k)*Catalan(n-k).

A078391 0 LD Triangle read by rows: T(n,k) = Catalan(k)*Catalan(n-k).

A078391 0 SC Triangle read by rows: T(n,k) = Catalan(k)*Catalan(n-k).

A078391 0 SD Triangle read by rows: T(n,k) = Catalan(k)*Catalan(n-k).

A078817 0 LD Table by antidiagonals giving variants on Catalan sequence: T(n,k)=C(2n,n)*C(2k,k)*(2k+1)/(n+k+1).

A078920 0 FC Upper triangle of Catalan Number Wall.

A080935 0 LD Triangle read by rows of number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2n steps with all values less than or equal to k.

A082635 0 FC Square array read by antidiagonals: degree of the K(2,p)^q variety.

A085180 0 FC Array A(x,y) giving the position of the y-th x in A007001 listed antidiagonalwise as A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), A(1,3), ...

A085843 0 FC Triangle T(n, k) read by rows; given by [1, 1, 1, 1, 1, ...] DELTA [1, 1, 2, 5, 14, 42, 132, 429, 1430, ...] (A000108) where DELTA is Deléham's operator defined in A084938.

A085853 0 FC Triangle T(n, k) read by rows; given by [1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 0, 2, 0, 2, 0, 3, 0, 2, 0, 4, 0, 2, 0, ...] (A000005 interspersed with 0's) where DELTA is Deléham's operator defined in A084938.

A085880 0 FC Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).

A085880 0 LD Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).

A086610 0 LD Triangle of coefficients, read by rows, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x) - x^2/(1-x)^2 + xy*f(x,y)^2.

A086612 0 LD Triangle of coefficients, read by rows, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = (1+x) - x^2*(1+x)^2 + xy*f(x,y)^2.

A086614 0 LD Triangle of coefficients, read by rows, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x)^2 + xy*f(x,y)^2.

A086636 0 FC Triangle of coefficients, read by rows, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = (3-sqrt(1-4x))/2 + xy*f(x,y)^3.

A086810 0 LD Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.

A088617 0 LD Triangle T(n,k) (n>=0, k=0..n) read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1).

A089434 0 LD Triangle read by rows: T(n,k) (n >=2, k >=0) is the number of non-crossing connected graphs on n nodes on a circle, having k interior faces. Rows are indexed 2,3,4,...; columns are indexed 0,1,2,....

A089435 0 LD Triangle read by rows: T(n,k) (n >=2, k >=0) is the number of non-crossing connected graphs on n nodes on a circle, having k triangles. Rows are indexed 2,3,4,...; columns are indexed 0,1,2,....

A090182 0 SC Triangle T(n,k), 0<=k<=n, composed of k-Catalan numbers.

A090285 0 FC Triangle T(n,k), 0<=k<=n, read by rows, defined by : T(n,k)=0 if k>n, T(n,0) = A000108(n); T(n+1,k)= Sum_{j=0..n} T(n-j,k-1)*binomial(2j+1,j+1).

A090299 0 FC Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.

A090985 0 LD Triangle read by rows: T(n,k) = number of dissections of a convex n-gon by nonintersecting diagonals, having exactly k triangles (n>=2, k>=0).

A091378 0 SC Triangle read by rows: T(m,n) = number of weak factorization systems (trivial Quillen model structures) on the poset of order-preserving maps from [m] to [n+1] (where [m] denotes the total order on m objects), viewed as a category.

A091378 0 SD Triangle read by rows: T(m,n) = number of weak factorization systems (trivial Quillen model structures) on the poset of order-preserving maps from [m] to [n+1] (where [m] denotes the total order on m objects), viewed as a category.

A092450 0 SC Triangle read by rows: T(m,n) = number of weak factorization systems (trivial Quillen model structures) on the product category [m]x[n], where [m] denotes the total order on m objects, viewed as a category.

A092450 0 SD Triangle read by rows: T(m,n) = number of weak factorization systems (trivial Quillen model structures) on the product category [m]x[n], where [m] denotes the total order on m objects, viewed as a category.

A092583 0 LD Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding the 123-pattern is equal to k.

A094236 0 FC Triangle read by rows: T(n,k) is the number of standard tableaux of shape (n,n,k) (0<=k<=n).

A094385 0 LD Another version of triangular array in A062991 unsigned and transposed : triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.

A094385 0 SC Another version of triangular array in A062991 unsigned and transposed : triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.

A094456 0 LD Triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.

A095801 0 FC Square of Narayana triangle A001263: View A001263 as a lower triangular matrix. Then the square of that matrix is also lower triangular. Sequence gives this lower triangle, read by rows.

A098273 0 FC Array by antidiagonals: Number of planar lattice walks of length 3n+2k starting at (0,0) and ending at (k,0), remaining in the first quadrant and using only NE,W,S steps.

A098474 0 LD Triangle read by rows, T(n,k) = C(n,k)*C(2*k,k)/(k+1), n>=0, 0<=k<=n.

A098509 0 FC Denominators of the inverse of a Catalan scaled binomial matrix.

A098509 0 LD Denominators of the inverse of a Catalan scaled binomial matrix.

A098977 0 SC Triangle read by rows: counts ordered trees by number of edges and position of first edge that terminates at a vertex of outdegree 1.

A099039 0 SC Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.

A101282 0 FC Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k valleys.

A101975 0 FC Triangle read by rows: number of Dyck paths of semilength n with k peaks after the first return (0<= k <n).

A104710 0 FC Triangle read by rows: reversed partial sums of Narayana triangle rows.

A104978 0 FC Triangle where g.f. satisfies: A(x,y) = 1 + x*A(x,y)^2 + x*y*A(x,y)^3, read by rows.

A105556 0 SC Triangle, read by rows, such that column n equals the row sums of A001263^n, which is the n-th matrix power of the Narayana triangle A001263, for n>=0.

A105848 0 FC Binomial transform of number triangle A105632.

A106270 0 FC Inverse of number triangle A106268.

A106270 0 SC Inverse of number triangle A106268.

A106534 0 LD Sum array of Catalan numbers A000108 read by antidiagonals.

A106566 0 SC Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, . . . ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, . . . ] where DELTA is the operator defined in A084938.

A107111 0 LD Number array whose rows are the series reversions of x(1-x)/(1+x)^k, read by antidiagonals.

A107842 0 FC A number triangle of lattice walks.

A108198 0 LD Triangle read by rows: T(n,k)=binomial(2k+2,k+1)*binomial(n,k)/(k+2) (0<=k<=n).

A108410 0 FC Triangle T(n,k) read by rows: number of 12312-avoiding matchings on [2n] with exactly k crossings (n >= 1, 0 <= k <= n-1 ).

A108410 0 LD Triangle T(n,k) read by rows: number of 12312-avoiding matchings on [2n] with exactly k crossings (n >= 1, 0 <= k <= n-1 ).

A108426 0 LD Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k peaks of the form Ud.

A108767 0 LD Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(1,1), d=(1,-2) and have k peaks (i.e. ud's).

A109450 0 LD Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.

A110488 0 FC A number triangle based on the Catalan numbers.

A110488 0 SC A number triangle based on the Catalan numbers.

A110857 0 SC Table T(n,k), n>=0, k>=0, product M*M^(T) where M is the lower triangular matrix in A106566 and M^(T) denotes the transpose matrix of M, read by antidiagonals.

A110857 0 SD Table T(n,k), n>=0, k>=0, product M*M^(T) where M is the lower triangular matrix in A106566 and M^(T) denotes the transpose matrix of M, read by antidiagonals.

A112338 0 SD Triangle read by rows, generated from A001263.

A114596 0 LD Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having abscissa of first return equal to 2k (2<=k<=n). A hill in a Dyck path is a peak at level 1.

A114715 0 FC Number A(n,m) of linear extensions of a 2 X n X m lattice; square array A(n,m), n>=1, m>=1, read by antidiagonals.

A114715 0 LD Number A(n,m) of linear extensions of a 2 X n X m lattice; square array A(n,m), n>=1, m>=1, read by antidiagonals.

A115126 0 LD First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).

A115126 0 SD First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).

A115179 0 LD Expansion of c(x*y(1-x)), c(x) the g.f. of A000108.

A115253 0 FC "Correlation triangle" for Catalan numbers.

A115253 0 LD "Correlation triangle" for Catalan numbers.

A116925 1 SD Triangle read by rows: row n (n>=0) consists of the elements g(i, n-i) (0 <= i <= n), where g(r,s) = 1 + Sum_{k=1..r} Product_{i=0..k-1} binomial(r+s-1,s+i) / binomial(r+s-1,i).

A117434 0 LD Expansion of c(x*y(1+x)), c(x) the g.f. of A000108.

A118964 0 FC Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that have k double rises above the x-axis (n>=1,k>=0). (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1); a double rise in a Grand Dyck path is an occurrence of uu in the path.)

A120406 0 FC Triangle read by rows: related to series expansion of the square root of 2 linear factors.

A120406 0 LD Triangle read by rows: related to series expansion of the square root of 2 linear factors.

A120986 0 FC Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n>=0, k>=0). A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child.

A120988 0 FC Triangle read by rows: T(n,k) is the number of binary trees with n edges and such that the first leaf in the preorder traversal is at level k (1<=k<=n). A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.

A122890 0 LD Triangle, read by rows, where the g.f. of row n divided by (1-x)^n yields the g.f. of column n in the triangle A122888, for n>=1.

A123352 0 SD Triangle read by rows, giving Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).

A124644 0 FC Mirror image of A098474 formatted as a triangular array.

A125177 0 FC Triangle read by rows: T(n,0)=C(2n,n)/(n+1) for n>=0; T(0,k)=0 for k>=1; T(n,k)=T(n-1,k)+T(n-1,k-1) for n>=1, k>=1.

A125311 0 LD Array giving number of (k,2)-noncrossing partitions of n, read by antidiagonals.

A126181 0 FC Triangle read by rows, T(n,k) = C(n,k)*Catalan(n-k+1), n>=0, 0<=k<=n.

A126216 0 FC Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n>=1; 0<=k<=n-1).

A127160 0 LD Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,2,3,4,5,6,...] DELTA [1,1,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938.

A127535 0 LD Triangle read by rows: T(n,k) is the number of even trees with 2n edges and jump-length equal to k (0<=k<=n-1). An even tree is an ordered tree in which each vertex has an even outdegree. In the preorder traversal of an ordered tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given ordered tree is called the jump-length.

A127767 0 LD Inverse of number triangle A(n,k)=if(k<=n,if(n<=2k,1/C(n),0),0), C(n)=A000108(n).

A127767 0 SD Inverse of number triangle A(n,k)=if(k<=n,if(n<=2k,1/C(n),0),0), C(n)=A000108(n).

A128567 0 FC Matrix square of Parker's partition triangle (A047812) read by rows.

A128899 0 SC Riordan array (1,(1-2x-sqrt(1-4x))/(2x)) .

A129159 0 LD Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having abscissa of the first return to the x-axis equal to 2k (1<=k<=n). A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.

A130020 0 SD Triangle T(n,k), 0<=k<=n, read by rows given by [1,0,0,0,0,0,0,...] DELTA [0,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938 .

A130513 0 FC Subtriangle of triangle in A051168: remove central column of A051168 and all columns to the right; now read by upwards diagonals.

A131427 0 LD A000108(n) preceded by n zeros.

A131429 0 FC (A000012 * A131427) + (A131427 * A000012) - A000012.

A131429 0 SC (A000012 * A131427) + (A131427 * A000012) - A000012.

A132808 0 FC A001263 * A103451 as infinite lower triangular matrices.

A133336 0 FC Triangle T(n,k), 0<=k<=n, read by rows, given by [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938 .

A134634 0 FC Triangle formed by Pascal's rule with left borders = A000108.

A134634 0 LD Triangle formed by Pascal's rule with left borders = A000108.

A135573 0 LD Array T(n,m) of super ballot numbers read along diagonals.

A135573 0 SC Array T(n,m) of super ballot numbers read along diagonals.

A135722 0 LD A000012 * A122890.

A135723 0 LD A122890 + A000012 - I, I = Identity matrix.

A136536 0 FC A001263 * A128064 * A000012 as infinite lower triangular matrices.

A137211 0 SC Generalized or s-Catalan numbers.

A140714 0 LD Triangle read by rows: T(n,k) is the number of white corners of rank k in all 321-avoiding permutations of {1,2,...,n} (n>=2, 0<=k<=n-2; for definitions see the Eriksson-Linusson references).

A141058 0 LD Pats by first entry.

A141058 0 SC Pats by first entry.

A141811 0 LD Partial Catalan numbers: triangle read by rows n = 1, 2, 3, ... and columns k = 0, 1, ..., n-1.

A145034 0 LD T(n,k) is the number of order-decreasing and order-preserving partial transformations (of an n-chain) of width (width(alpha) = |Dom(alpha)|) and waist (waist(alpha) = max(Im(alpha))) both equal to k.

A145879 0 SC 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).

A145890 0 FC Triangle read by rows: T(n,k)=B(k)C(n-k), where B(j) is the central binomial coefficient binom(2j,j) (A000984) and C(j) is the Catalan number binom(2j,j)/(j+1) (A000108); 0<=k<=n).

A146305 0 LD Array T(n,m) = 2(2m+3)!(4n+2m+1)!/(m!(m+2)!n!(3n+2m+3)!) read by antidiagonals.

A147294 0 FC Eigentriangle, row sums = A125275

A155586 0 SC A modified Catalan sequence array.

A157491 0 LD A050165*A130595 as infinite lower triangular matrices.

A157491 0 SC A050165*A130595 as infinite lower triangular matrices.

A157513 0 FC Triangle of numbers of walks in the quarter-plane, of length 2n beginning and ending at the origin using steps {(1,1), (1,0), (-1,0), (-1,-1)} (Gessel steps) arranged according to the number of times the steps (1,1) and (-1,-1) occur.

A157513 0 LD Triangle of numbers of walks in the quarter-plane, of length 2n beginning and ending at the origin using steps {(1,1), (1,0), (-1,0), (-1,-1)} (Gessel steps) arranged according to the number of times the steps (1,1) and (-1,-1) occur.

A158825 0 LD Square array of coefficients in the successive iterations of x*C(x) = (1-sqrt(1-4*x))/2 where C(x) is the g.f. of the Catalan numbers (A000108); read by antidiagonals.

A158830 0 FC Triangle, read by rows n>=1, where row n is the n-th differences of column n of array A158825, where the g.f. of row n of A158825 is the n-th iteration of x*Catalan(x).

A162382 0 FC Triangle, read by rows, defined by: T(n,k) = 1/((k+1)n-1) binomial((k+1)n-1,n) for n,k>0.

A163946 0 FC Triangle read by rows, A033184 * A091768 (diagonalized as an infinite lower triangular matrix).

A163946 0 SC Triangle read by rows, A033184 * A091768 (diagonalized as an infinite lower triangular matrix).

A167685 0 FC Triangle read by rows given by [1,1,1,1,1,1,1,1,1,1,...] DELTA [1,1,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

A168256 0 FC Triangle read by rows: Catalan number C(n) repeated n+1 times.

A168256 0 LD Triangle read by rows: Catalan number C(n) repeated n+1 times.

A168256 0 SC Triangle read by rows: Catalan number C(n) repeated n+1 times.

A168256 0 SD Triangle read by rows: Catalan number C(n) repeated n+1 times.

A168391 0 LD 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}]

A169589 0 FC A number triangle with repeated columns of triangle in A039599.

A169589 0 SC A number triangle with repeated columns of triangle in A039599.

A171567 0 FC Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A168491.

A171567 0 SC Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A168491.

A172380 0 SC Eigentriangle of Catalan triangle A033184.

A172381 0 FC Triangle whose inverse has production matrix with general term (-1)^(n-k+1)*C(k+1, n-k+1).

A172417 0 FC n*Catalan number(n+1) triangle.

A172417 0 LD n*Catalan number(n+1) triangle.

A172417 0 SC n*Catalan number(n+1) triangle.

A172417 0 SD n*Catalan number(n+1) triangle.

A173050 0 SC Triangle, read by rows, given by [0,1,1,1,1,1,1,1,...] DELTA [1,0,1,0,2,0,3,0,4,0,5,0,6,0,...] where DELTA is the operator defined in A084938.

A177267 0 FC Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having genus k (see first comment for definition of genus).

A178518 0 FC Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} having genus 0 and such that p(1)=k (see first comment for definition of genus).

A178518 0 LD Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} having genus 0 and such that p(1)=k (see first comment for definition of genus).

A178518 0 SC Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} having genus 0 and such that p(1)=k (see first comment for definition of genus).

A181196 0 SC T(n,k) = number of n X k matrices containing a permutation of 1..n*k in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

A181645 0 FC Triangle Id-(xc(x),xc(x)), c(x) the g.f. of the Catalan numbers A000108.

A181645 0 SC Triangle Id-(xc(x),xc(x)), c(x) the g.f. of the Catalan numbers A000108.

A182534 0 FC Array read by antidiagonals: coefficient of the Euler-Mascheroni constant in below expression.

A185209 0 FC Triangle read by rows: T(n,k) is the number of indecomposable (connected) permutations of {1,2,...,n} having genus k (see first comment for definition of genus).

A185249 1 SD Triangle read by rows: Table III.5 of Myriam de Sainte-Catherine's 1983 thesis.

A189675 0 FC Composition of Catalan and Fibonacci numbers.

A202992 0 FC 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.

A204057 0 SD Triangle derived from an array of f(x), Narayana polynomials.

A210658 0 LD Triangle of partial sums of Catalan numbers.

A211400 0 SC Rectangular array, read by upward diagonals: T(n,m) is the number of Young tableaux that can be realized as the ranks of the outer sums a_i + b_j where a = (a_1, ... a_n) and b = (b_1, ... b_m) are real monotone vectors in general position (all sums different).

A211400 0 SD Rectangular array, read by upward diagonals: T(n,m) is the number of Young tableaux that can be realized as the ranks of the outer sums a_i + b_j where a = (a_1, ... a_n) and b = (b_1, ... b_m) are real monotone vectors in general position (all sums different).

A212363 0 SD Number A(n,k) of Dyck n-paths all of whose ascents and descents have lengths equal to 1+k*m (m>=0); square array A(n,k), n>=0, k>=0, read by antidiagonals.

A212382 0 SD Number A(n,k) of Dyck n-paths all of whose ascents have lengths equal to 1+k*m (m>=0); square array A(n,k), n>=0, k>=0, read by antidiagonals.

A213946 0 LD A Catalan triangle read by rows, derived from the INVERT transform of initial segments of the Catalan numbers A000108.

A214015 1 SD Number of permutations A(n,k) in S_n with longest increasing subsequence of length <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A214722 0 LD Number A(n,k) of solid standard Young tableaux of shape [[{n}^k],[n]]; square array A(n,k), n>=0, k>=1, read by antidiagonals.

A214775 0 FC Number T(n,k) of solid standard Young tableaux of shape [[n,k],[n-k]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A214775 0 LD Number T(n,k) of solid standard Young tableaux of shape [[n,k],[n-k]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A214776 0 SD Number A(n,k) of standard Young tableaux of shape [n*k,n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A224652 0 SC Triangle read by rows: T(n,k) is the number of permutations of n elements with k the (smallest) header (first element) of the longest descending subsequence.

A227159 0 LD Triangle read by rows: number of 321-avoiding ordered set partitions of [n] into k blocks, n>=1, 1<=k<=n.

A234950 0 FC Borel's triangle read by rows: T(n,k) = Sum_{s=k..n} binomial(s,k)*C(n,s), where C(n,s) is an entry in Catalan's triangle A009766.

A234950 0 LD Borel's triangle read by rows: T(n,k) = Sum_{s=k..n} binomial(s,k)*C(n,s), where C(n,s) is an entry in Catalan's triangle A009766.

A236843 0 FC Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

A237018 0 SD Number A(n,k) of partitions of the k-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A241262 0 FC Array t(n,k) = binomial(n*k, n+1)/n, where n >= 1 and k >= 2, read by ascending antidiagonals.

A243631 0 SD Square array of Narayana polynomials N_n evaluated at the integers, A(n,k) = N_n(k), n>=0, k>=0, read by antidiagonals.

A243660 0 LD Triangle read by rows: the x = 1+q Narayana triangle at m=2.

A245559 0 SD Triangle read by rows: entries on or below the main diagonal in A245558.

A246322 0 LD Triangle read by rows: T(n,k) = number of neutral planar lambda terms of size n with k free variables (n >= 0, 1 <= k <= n+1).

A246323 0 LD Triangle read by rows: T(n,k) = number of normal planar lambda terms of size n with k free variables (n >= 1, 1 <= k <= n).

A247507 0 LD Square array read by ascending antidiagonals, n>=0, k>=0. Row n is the expansion of (1-n*x-sqrt(n^2*x^2-2*n*x-4*x+1))/(2*x).

A247582 0 FC Triangle, read by rows, T(n,k) = (k+1)*Sum_{i=0..n-k} C(k+2*i,i)*C(n-i-1,n-k-i)/(k+i+1).

A253180 0 LD Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types that are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A253180 0 SC Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types that are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A254632 0 LD Triangle read by rows, T(n, k) = 4^n*[x^k]hypergeometric([3/2, -n], [3], -x), n>=0, 0<=k<=n.

A255982 0 SC Number T(n,k) of partitions of the k-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A256061 0 SC Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A256117 0 LD Number T(n,k) of length 2n words such that all letters of the k-ary alphabet occur at least once and are introduced in ascending order and which can be built by repeatedly inserting doublets into the initially empty word; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A256640 0 FC Triangle read by rows: T(n,k) = Sum_{i=n-k..n} C(k-1,n-i)*C(i,n-k)*C(2*i,i)/(i+1).

A257813 0 SD G.f. satisfies: A(x,y) = 1-x + y*x + Series_Reversion( x/A(x,y)^2 ).

A258222 0 LD A(n,k) is the sum over all Dyck paths of semilength n of products over all peaks p of (k*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A258223 0 FC T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258222(n,i); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A259101 0 FC Square array read by antidiagonals arising in the enumeration of corners.

A259101 0 LD Square array read by antidiagonals arising in the enumeration of corners.

A259332 0 LD Triangle read by rows: T(n,k) = number of column-convex polyominoes with perimeter n and k columns (1 <= k <= n).

A259356 0 SC Triangle T(n,k) read by rows: T(n,k) is the number of closed lambda-terms of size n with size 0 for the variables and k abstractions.

A261665 0 LD Triangle read by rows: T(n,k) = number of k-classes of permutations of n letters avoiding the pattern 132 (n>=1, 0 <= k <= n-1)

A261665 0 SD Triangle read by rows: T(n,k) = number of k-classes of permutations of n letters avoiding the pattern 132 (n>=1, 0 <= k <= n-1)

A263791 0 SD Number of permutations of [n] avoiding the generalized patterns 1(k+2)-(u_1+1)-...-(u_k+1) for all permutations u of [k].

A267998 0 SD Table array: T(n,k) is Ann_k(2*n+k,k) where Ann_k(n,m) is the number of annular non-crossing matchings of type (n, m) with precisely k cross-cuts.

A268652 0 LD G.f. satisfies: A(x,y) = 1 + x*y*A(x,y+1)^2.

A269920 0 FC Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces of an orientable surface of genus 0.

A269920 0 LD Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces of an orientable surface of genus 0.

A271025 0 FC Array read by anti-diagonals: T(i,j) is the i-th binomial transform of the Catalan sequence (A000108) evaluated at j.

A271825 0 FC Triangle read by rows: T(n,m) = (-1)^(n-m-1)*m*binomial(2*n-3*m-1,n-m-1)/(n-m), T(n,n)=1.

A271825 0 SC Triangle read by rows: T(n,m) = (-1)^(n-m-1)*m*binomial(2*n-3*m-1,n-m-1)/(n-m), T(n,n)=1.

A271875 0 FC Triangle T(n,m) = Sum_{k=1..n-m}(k*(-1)^k*binomial(m+k-1,k)*binomial(2*(n-m),n-m-k))/(n-m), T(n,n)=1.

A271875 0 SC Triangle T(n,m) = Sum_{k=1..n-m}(k*(-1)^k*binomial(m+k-1,k)*binomial(2*(n-m),n-m-k))/(n-m), T(n,n)=1.


near misses or look alikes

Catalan Numbers:

1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, ...


A120493 0 FC Triangle T(n,k) read by rows ; multiply row n of Pascal's triangle (A007318) by A024175(n). {1, 1, 2, 5, 14, 42, 132, 428, 1416}

A120493 0 LD Triangle T(n,k) read by rows ; multiply row n of Pascal's triangle (A007318) by A024175(n). {1, 1, 2, 5, 14, 42, 132, 428, 1416}

A122881 1 SD Triangle read by rows: number of Catalan paths of 2n steps of all values less than or equal to m. {1, 1, 2, 5, 14, 42, 132, 429, 1429}

A165189 0 SC Partial sums of partial sums of (A001840 interleaved with zeros). {5, 14, 42, 132, 390, 1040, 2550, 5852}

A180682 1 SD a(n) is the largest path count within the (right-aligned Ferrers plots of the) partitions of n. {1, 2, 5, 14, 42, 132, 429, 1430, 5096}

A239880 3 SD Number of strict partitions of n having an ordering in which no parts of equal parity are juxtaposed and the first and last terms have the same parity. {0, 1, 1, 5, 14, 42, 132, 394, 1262}