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# Triangles Row Sum Power2

## Contents

### Powers of 2

A007318 Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0 <= k <= n.

A027935 Triangular array T read by rows: T(n,k)=t(n,2k), t given by A027926; 0<=k<=n, n >= 0.

A027948 Triangular array T read by rows: T(n,k)=t(n,2k+1) for 0<=k<=[ (2n-1)/2 ], T(n,n)=1, t given by A027926, n >= 0.

A028326 Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).

A037088 Triangle read by rows: T(n,k) is number of numbers x, 2^n <= x < 2^(n+1), with k prime factors (counted with multiplicity).

A046688 Antidiagonals of square array in which k-th row (k>0) is an A.P. of difference 2^(k-1).

A048003 Triangular array T read by rows: T(h,k) = number of binary words of length h and maximal runlength k.

A048004 Triangular array read by rows: T(n,k) = number of binary vectors of length n whose longest run of consecutive 1's has length k, for n >= 0, 0 <= k <= n.

A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum[binomial[k,m]binomial[n-k,k-m],{k,0,n} ], (see program line).

A053633 Triangular array T(n,k) giving coefficients in expansion of Product_{j=1..n} (1+x^j) mod x^(n+1)-1.

A057728 A triangular table of decreasing powers of two (with first column all ones).

A061554 Square table read by antidiagonals: a(n,k) = binomial(n+k, floor(k/2)).

A071919 Number of monotone nondecreasing functions [n]->[m] for n>=0, m>=0, read by antidiagonals.

A076261 Triangle T(n,k) (n >= 2, 1 <= k <= n-1) read by rows, where T(n,k) is the number of words of length n in the free group on two generators that require exactly k multiplications for their formation.

A080521 Triangle read by rows in which the n-th row contains n distinct numbers whose sum is the smallest n-th power, i.e. 2^n.

A087698 Triangle read by rows, giving T(n,k) = maximum number of examples (Boolean inputs) at Hamming distance 2 for symmetric Boolean functions that can have different outputs.

A089818 T(n,k) = number of subsets of {1,..., n} containing exactly k primes, triangle read by rows, 0<=k<n.

A089886 T(n,k) = number of subsets of {1,..., n} containing exactly k squares, triangle read by rows, 0<=k<n.

A091613 Triangle: T(n,k) = number of compositions (ordered partitions) of n such that some part is repeated consecutively k times and no part is repeated consecutively more than k times.

A092479 T(n,k) = number of numbers <= 2^n having exactly k prime factors (with repetitions), 0<=k<=n, triangle read by rows.

A093560 (3,1) Pascal triangle.

A094495 Table of binomial coefficients mod m^2, read by rows: T(m, n) = binomial(m, n) mod m^2.

A095660 Pascal (1,3) triangle.

A095704 Triangle read by rows giving coefficients of the trigonometric expansion of sin(n*x).

A097805 Riordan array (1, 1/(1-x)) read by rows.

A098157 Triangle T(n,k) with diagonals T(n,n-k)=binomial(n+1,2k).

A098158 Triangle T(n,k) with diagonals T(n,n-k)=binomial(n,2k).

A100257 Triangle of expansions of 2^(k-1)*x^k in terms of T(n,x), in descending degrees n of T, with T the Chebyshev polynomials.

A100749 Triangle read by rows: T(n,k)=number of 231- and 312-avoiding permutations of [n] having k fixed points.

A103284 Triangle, read by rows, where row n+1 is formed by sorting, in ascending order, the result of the convolution of row n with {1,1}.

A105147 Triangular array read by rows: T(n,k) = number of compositions of n having smallest part equal to k.

A105422 Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 1.

A106356 Triangle T(n,k) 0<=k<n : Number of compositions of n with k adjacent equal parts.

A107430 Triangle read by rows: row n is row n of Pascal's triangle (A007318) sorted into increasing order.

A108086 Triangle, read by rows, where T(0,0) = 1, T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1); a signed version of Pascal's triangle.

A108723 Triangle read by rows: T(n,k) is number of permutations of [n] with ascending runs consisting of consecutive integers and having k fixed points.

A110555 Triangle of partial sums of alternating binomial coefficients: T(n,k) = Sum(binomial(n,k)*(-1)^k: 0<=k<=n).

A110971 Triangle T(n,k) (n>=2, 1<=k<=n-1) read by rows: row n gives epispectrum of a path P_n (see reference for precise definition).

A117440 A cyclically signed version of Pascal's triangle.

A118400 Triangle T, read by rows, where all columns of T are different and yet all columns of the matrix square T^2 (A118401) are equal; a signed version of triangle A087698.

A118433 Self-inverse triangle H, read by rows; a nontrivial matrix square-root of identity: H^2 = I, where H(n,k) = C(n,k)*(-1)^([(n+1)/2]-[k/2]+n-k) for n>=k>=0.

A119458 Triangle read by rows: T(n,k) is the number of circular binary words of length n having k occurrences of 00 (0<=k<=n).

A119467 A masked Pascal triangle.

A119900 Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs (0<=k<=n; for example, the binary word 1/0/01/01/1/1/01 has 7 strictly increasing runs).

A120643 Table T(n,k) = number of fractal initial sequences (where new values are successive integers) of length n whose last term is k.

A120933 Triangle read by rows: T(n,k) is the number of binary words of length n for which the length of the maximal leading nondecreasing subword is k (1<=k<=n).

A122950 Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

A124846 Triangle read by rows: T(n,k)=[2-(-1)^k]binom(n,k) (0<=k<=n).

A125104 Triangle read by rows counting compositions (ordered partitions) by minimal part size.

A127159 Triangle T(n,k) with T(n,k)=A061554(n,k)+A107430(n,k).

A130123 Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0,...).

A130595 Triangle read by rows: lower triangular matrix which is inverse to Pascal's triangle (A007318) regarded as a lower triangular matrix.

A131047 (1/2) * ((A007318 - A007318^(-1)).

A131127 Table read by rows: 2*A007318(n,m) - (-1)^(n+m)*A097806(n,m).

A131909 Triangle, read by rows, where T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n>=k>1, with T(0,0)=1 and T(n,0) = T(n+1,1) = T(n-1,n-1) for n>0.

A133114 A000012 * A007318 * A133080.

A134058 Triangle read by rows, T(n,k) = 2*binomial(n,k) if k>0, (0<=k<=n), left column = (1,2,2,2,...).

A134309 Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).

A134388 A generalized Riordan array related to Hankel and Toeplitz+Hankel transforms.

A135225 Pascal's triangle A007318 augmented with a leftmost border column of 1's.

A135392 A triangular sequence from a general proportionality to modular function polynomial triangular function.

A140643 First differences of A140642.

A140993 Triangle read by rows: recurrence G(n,k): G(n, n)=G(n+1, 1)=1, G(n+2, 2)=2, G(n+3, k)=G(n+1, k-1)+G(n+1, k-2)+G(n+2, k-1) for k:=2..(n+2), 0<=k<=n.

A140994 Triangle read by rows: recurrence G(n,k): G(n, n) = G(n+1, 0)=1, G(n+2, 1)=2, G(n+3, 2)=4, G(n+4, k) = G(n+1, k-2) + G(n+1, k-3) + G(n+2, k-2) + G(n+3, k-1) for k:=3..(n+3), 0<=k<=n.

A140995 Triangle read by rows: recurrence G(n,k): G(n, n)=G(n+1, 0)=1, G(n+2, 1)=2, G(n+3, 2)=4, G(n+4, 3)=8, G(n+5, k)=G(n+1, k-3)+G(n+1, k-4)+G(n+2, k-3)+G(n+3, k-2)+ G(k+4, k-1), for k:=5..(n+5).

A140996 Triangle read by rows: recurrence G(n,k): G(n, 0)=G(n+1, n+1)=1, G(n+2, n+1)=2, G(n+3, n+2)=4, G(n+4, n+3)=8, G(n+5, k)=G(n+1, k-1)+G(n+1, k)+G(n+2, k)+G(n+3, k)+ G(k+4, k), for k:=1..(n+1).

A140997 Triangle G(n,k) read by rows: G(n, 0)=G(n, n)=1, G(n, n-1)=2, G(n, n-2)=4. Recurrence G(n, k)=G(n-1, k)+G(n-2, k)+G(n-3, k)+ G(n-3, k-1), for k:=1..n-3.

A140998 Triangle read by rows: recurrence G(n,k): G(n, 0)=G(n+1,n+1)=1, G(n+2, n+1)=2, G(n+3, k)=G(n+1,k-1)+G(n+1, k)+G(n+2, k), for k:=1..(n+1).

A141020 Pascal-like triangle with index of asymmetry y=4 and index of obliqueness z=0.

A141021 Pascal-like triangle with index of asymmetry y=4 and index of obliqueness z=1.

A141665 A signed half of Pascal's triangle A007318: p(x,n) = (1+I*x)^n; t(n,m) = real part of coefficients(p(x,n)).

A144157 Eigentriangle, row sums = A011782: (1, 1, 2, 4, 8, 16,...).

A144225 Bordered Pascal's triangle in rectangular format.

A152538 Triangle read by rows, A027293 * (A152537 * 0^(n-k))

A152568 Triangle T(n,k) read by rows: T(n,n) = -1, T(n,1) = 2^(n-1), T(n,k) = -2^(n-k), 1<=k<n.

A154312 Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,0,0,...] DELTA [2,-1/2,-1/2,2,0,0,0,0,0,0,0 ...] where DELTA is the operator defined in A084938 .

A154926 Signed version of Pascal's triangle. Diagonal positive, rest negative.

A155038 Triangle read by rows: T(n,k) is the number of compositions of n with first part k.

A155997 Triangle read by rows: t0(n,m)=(Binomial[n, m] + (-1)^m*Binomial[n, m])/2; t(n,m)=t(n,m)+t0(n,n-m).

A155998 Triangle read by rows: t0(n,m)=(Binomial[n, m] -(-1)^m*Binomial[n, m])/2; t(n,m)=t(n,m)+t0(n,n-m).

A159255 Expansion of (1-x+x^2)*(1+x)^n .

A159853 Riordan array ((1-2*x+2*x^2)/(1-x),x/(1-x)).

A159854 Riordan array (1-x,x/(1-x)).

A159916 Triangle T[m,n] = number of subsets of {1,...,m} with n elements having an odd sum, 1 <= n <= m.

A162315 Triangular array 2*P - P^-1, where P is Pascal's triangle A007318.

A162590 Polynomials with e.g.f. exp(x*t)/csch(t), triangle of coefficients read by rows.

A166065 Triangle, read by rows, given by [0,1,1,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

A169940 Consider the 2^(n-1) monic polynomials f(x) with coefficients 0 or 1, degree n and f(0)=1. Sequence gives triangle read by rows, in which T(n,k) (n>=1) is the number of such polynomials of thickness k (2 <= k <= n+1).

A169941 Triangle read by rows: A169940 with rows reversed.

A169945 Consider the 2^(n+1) polynomials f(x) with coefficients 0 or 1 and degree <= n. Sequence gives triangle read by rows, in which T(n,k) (n>=0) is the number of such polynomials of thickness k (0 <= k <= n+1).

A169946 Triangle read by rows: A169945 with rows reversed.

A169950 Consider the 2^n monic polynomials f(x) with coefficients 0 or 1 and degree n. Sequence gives triangle read by rows, in which T(n,k) (n>=0) is the number of such polynomials of thickness k (1 <= k <= n+1).

A169951 Triangle read by rows: A169950 with rows reversed.

A177767 A triangle of Polynomial Coefficients whose row sums are A000225:p(x,n)=If[n==0,1,x*(1 + x)^(n - 1) - 1]

A185778 Second weight array of Pascal's triangle (formatted as a rectangle), by antidiagonals.

A187801 Pascal's triangle construction method applied to {1,1,2} as an initial term.

A193554 Triangle read by rows: first column: top entry is 1, then powers of 2; rest of triangle is Pascal's triangle A007318.

A193787 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+1)^n and q(n,x)=1+x^n.

A194288 Triangular array: g(n,k)=number of fractional parts (i*sqrt(2)) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n.

A194292 Triangular array: g(n,k)=number of fractional parts (i*sqrt(3)) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n.

A194296 Triangular array: g(n,k)=number of fractional parts (i*r) in interval [(k-1)/n, k/n], for 1<=i<=n, 1<=k<=2^n, r=(1+sqrt(5))/2, the golden ratio.

A194300 Triangular array: g(n,k)=number of fractional parts (i*r) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n, r=(1+sqrt(3))/2.

A194304 Triangular array: g(n,k)=number of fractional parts (i*sqrt(5)) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n.

A194308 Triangular array: g(n,k)=number of fractional parts (i*pi) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n.

A194312 Triangular array: g(n,k)=number of fractional parts (i*e) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n.

A194316 Triangular array: g(n,k)=number of fractional parts (i*sqrt(6)) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n.

A194320 Triangular array: g(n,k)=number of fractional parts (i*sqrt(8)) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n.

A194324 Triangular array: g(n,k)=number of fractional parts (i*sqrt(1/2)) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n.

A194328 Triangular array: g(n,k)=number of fractional parts (i*r) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n, r=2-sqrt(2).

A194332 Triangular array: g(n,k)=number of fractional parts (i*r) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n, r=2-sqrt(3).

A194336 Triangular array: g(n,k)=number of fractional parts (i*r) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n, r=2-tau, where tau=(1+sqrt(5))/2, the golden ratio.

A194340 Triangular array: g(n,k)=number of fractional parts (i*r) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n, r=3-sqrt(5).

A194344 Triangular array: g(n,k)=number of fractional parts (i*r) in interval [(k-1)/n, k/n], for 1<=i<=2^n, 1<=k<=n, r=3-e.

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

A202389 Triangle T(n,k), read by rows, given by (1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

A208342 Triangle of coefficients of polynomials u(n,x) jointly generated with A208343; see the Formula section.

A208343 Triangle of coefficients of polynomials v(n,x) jointly generated with A208342; see the Formula section.

A208891 Pascal's triangle matrix augmented with a right border of 1's.

A209599 Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

A209919 Triangle read by rows: T(n,k), 0 <= k <= n-1, = number of 2-divided binary sequences of length n which are 2-divisible in exactly k ways.

A210381 Triangle by rows, derived from the beheaded Pascal's triangle, A074909

A210868 Triangle of coefficients of polynomials u(n,x) jointly generated with A210869; see the Formula section.

A210869 Triangle of coefficients of polynomials v(n,x) jointly generated with A210868; see the Formula section.

A210873 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.

A213945 Triangle by rows, generated from aerated sequences of 1's.

A214258 Number T(n,k) of compositions of n where the difference between largest and smallest parts equals k; triangle T(n,k), n>=1, 0<=k<n, read by rows.

A216953 Triangle read by rows: T(n,k) (n>=1, 1<=k<=n) = number of binary sequences of length n with minimal period k.

A216954 Triangle read by rows: A216953/2.

A216955 Triangle read by rows: T(n,k) (n>=1, 1<=k<=n) = number of binary sequences of length n and curling number k.

A216956 Triangle read by rows: A216955/2.

A225084 Triangle read by rows: T(n,k) is the number of compositions of n with maximal up-step k; n>=1, 0<=k<n.

A232089 Table read by rows, which consist of 1 followed by 2^k, 0 <= k < n ; n = 0,1,2,3,...

A236076 A skewed version of triangular array A122075.

A238130 Triangle read by rows: T(n,k) is the number of compositions into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=n.

A238341 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with exactly k occurrences of the largest part, n>=0, 0<=k<=n.

A238342 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with exactly k occurrences of the smallest part, n>=0, 0<=k<=n.

A238343 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k descents, n>=0, 0<=k<=n.

A238345 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the first occurrence of a largest part, n>=1, 1<=k<=n.

A238346 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the last occurrence of a largest part, n>=1, 1<=k<=n.

A238347 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the first occurrence of a smallest part, n>=1, 1<=k<=n.

A238348 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the last occurrence of a smallest part, n>=1, 1<=k<=n.

A238349 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k parts p at position p (fixed points), n>=0, 0<=k<=n.

A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

A242447 Number T(n,k) of compositions of n in which the maximal multiplicity of parts equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A242451 Number T(n,k) of compositions of n in which the minimal multiplicity of parts equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

A265848 Pascal's triangle, right and left halves interchanged.

A269456 Triangular array T(n,k) read by rows: T(n,k) is the number of degree n monic polynomials in GF(2)[x] with exactly k factors in its unique factorization into irreducible polynomials.

A273496 Triangle read by rows: coefficients in the expansion cos(x)^n = (1/2)^n * Sum_{k=0..n} T(n,k) * cos(k*x).

### Powers of 4 , even powers of 2

A013610 Triangle of coefficients in expansion of (1+3*x)^n.

A024462 Triangle T(n,k) read by rows, arising in enumeration of catafusenes.

A027465 Cube of lower triangular normalized binomial matrix.

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

A038208 Triangle whose (i,j)-th entry is binomial(i,j)*2^i.

A067804 Triangle read by rows: T(n,k) = number of walks (each step +/-1) of length 2n which have a cumulative value of 0 last at step 2k.

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

A091042 Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.

A091044 One half of odd numbered entries of even numbered rows of Pascal's triangle A007318.

A103327 Triangle T(n, k) read by rows: binomial(2n+1, 2k+1).

A111418 Right-hand side of odd-numbered rows of Pascal's triangle.

A113187 Inverse of twin-prime related triangle A111125.

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

A122366 Triangle read by rows: T(n,k) = binomial(2*n+1,k), 0<=k<=n.

A124572 Triangle read by rows where the n-th row is the first row of M^n, with M the (n+1)-by-(n+1) matrix with (1,3,1,3,1,3...) on its main diagonal and (3,1,3,1,3,1...) on its superdiagonal.

A124573 Triangle read by rows where the n-th row is the first row of M^n, with M the (n+1)-by-(n+1) matrix with (3,1,3,1,3,1...) on its main diagonal and (1,3,1,3,1,3...) on its superdiagonal.

A125077 #4 in an infinite set of generalized Pascal's triangles with trigonometric properties.

A131049 (1/4) * (A007318^3 - A007318^(-1)).

A135299 Pascal's triangle, but the last element of the row is the sum of the all the previous terms.

A140714 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).

A158335 A triangle of matrix polynomials: m(n)=antisymmeticmatix(n).Transpose[antisymmeticmatix(n)].

A180063 Pascal-like triangle with trigonometric properties, row sums = powers of 4; generated from shifted columns of triangle A180062.

A201972 Triangle T(n,k), read by rows, given by (2,1/2,-1/2,0,0,0,0,0,0,0,...) DELTA (2,-1/2,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

A202396 Triangle T(n,k), read by rows, given by (2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

A210637 Triangle T(n,k), read by rows, given by (2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

A210804 Triangle of coefficients of polynomials v(n,x) jointly generated with A210803; see the Formula section.

A265644 Triangle read by rows: T(n,m) is the number of quaternary words of length n with m strictly increasing runs (0<=m<=n).

### odd powers of 2

A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

A055372 Invert transform of Pascal's triangle A007318.

A086645 Triangle read by rows: T(n; k) = Binomial(2*n; 2*k).

A103328 Triangle T(n, k) read by rows: binomial(2n, 2k+1).

A110552 A triangular array related to A077028 and distributing the values of A007582.

A112358 The following triangle is based on Pascal's triangle. The r-th term of the n-th row is sum of C(n,r) successive integers so that the sum of all the terms of the row is (2^n)*(2^n+1)/2, the 2^n -th triangular number. Sequence contains the triangle read by rows.

A127673 One half of even powers of 2*x in terms of Chebyshev's T-polynomials.

A136158 A136157^n * [1, 1, 0, 0, 0,..] generates rows of A136158.

A146766 A new symmetrical polynomial form to give a triangle sequence: p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 2)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].

A164705 T(n,k)=Binomial(2n-k,n)*2^(k-1) for n>=1,0<=k<=n

A164948 Fibonacci matrix read by anti-diagonals. (Inverse of A136158)

A193792 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+3)^n and q(n,x)=1+x^n.

A193793 Mirror of the triangle A193792.

A193794 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(3x+1)^n and q(n,x)=1+x^n.

A193795 Mirror of the triangle A193794.

A202395 Triangle T(n,k), read by rows, given by (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

### Powers of 8 , 2^(3n)

A013614 Triangle of coefficients in expansion of (1+7x)^n.

A013622 Triangle of coefficients in expansion of (3+5x)^n.

A027466 Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).

A038212 Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j)*6^j.

A038234 Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j)*4^j.

A038245 Triangle whose (i,j)-th entry is binomial(i,j)*5^(i-j)*3^j.

A038256 Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*2^j.

### 2^(3n+1)

A055374 Invert transform applied three times to Pascal's triangle A007318.

### 2^(3n+2)

none found yet.

### Powers of 16

A027467 Triangle whose (n,k)-th entry is binomial(n,k)*15^(n-k).

A038242 Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j)*12^j.

A038253 Triangle whose (i,j)-th entry is binomial(i,j)*5^(i-j)*11^j.

A038264 Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*10^j.

A038275 Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*9^j.

A038286 Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*8^j.

A038297 Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*7^j.

A038308 Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*6^j.

A038319 Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.

A038330 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.

### 2^(n*(n-1)/2)

A143543 Triangle read by rows: T(n,k) = number of labeled graphs on n nodes with k connected components, 1<=k<=n.

A156960 q-Carlitz-Al-Salam-Appell polynomial coefficients:q=2; p(x,n)=x*p[x, n - 1] - (1 - q^(n - 1))*q^(n - 2)*p[x, n - 2].

A198261 Triangular array read by rows T(n,k) is the number of simple labeled graphs on n nodes with exactly k isolated nodes, 0<=k<=n.

A228550 Triangular array read by rows: T(n,k) is the number of simple labeled graphs on n nodes in which each vertex has even degree and having exactly k components; n>=1, 1<=k<=n.

### 2^(n*(n+1))

A189898 Triangular array read by rows. T(n,k) is the number of weakly connected digraphs with n nodes having exactly k components. n >= 1, 1 <= k <= n.

A217580 Triangular array read by rows. T(n,k) is the number of labeled digraphs on n nodes with exactly k isolated nodes. 0<=k<=n.

### 2^(n^2)

A064230 Triangle T(n,k) = number of rational (0,1) matrices of rank k (n >= 0, 0 <= k <= n).

A064231 Triangle read by rows: T(n,k) = number of rational (+1,-1) matrices of rank k (n >= 1, 1 <= k <= n).

A191249 Triangular array T(n,k) read by rows: number of labeled relations of the n-set with exactly k connected components.

A217436 Triangular array read by rows. T(n,k) is the number of labeled relations on n elements with exactly k vertices of indegree and outdegree = 0.