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 A087127 This table shows the coefficients of combinatorial formulas needed for generating the sequential sums of p-th powers of triangular numbers. The p-th row (p>=1) contains a(i,p) for i=1 to 2*p-1, where a(i,p) satisfies Sum_{i=1..n} C(i+1,2)^p = 3 * C(n+2,3) * Sum_{i=1..2*p-1} a(i,p) * C(n-1,i-1)/(i+2). 38
 1, 1, 2, 1, 1, 8, 19, 18, 6, 1, 26, 163, 432, 564, 360, 90, 1, 80, 1135, 6354, 18078, 28800, 26100, 12600, 2520, 1, 242, 7291, 77400, 405060, 1210680, 2211570, 2520000, 1751400, 680400, 113400, 1, 728, 45199, 862218, 7667646, 38350080, 118848420 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS From Peter Bala, Mar 08 2018: (Start) The table entries T(n,k) are the coefficients when expressing the polynomial C(x+2,2)^p of degree 2*p in terms of falling factorials: C(x+2,2)^p = Sum_{k = 0..2*p} T(p,k)*C(x,k). It follows that Sum_{i = 0..n-1} C(i+2,2)^p = Sum_{k = 0..2*p} T(p,k)*C(n,k+1). The sum of the p-th powers of the triangular numbers is also given by Sum_{i = 0..n-1} C(i+2,2)^p = Sum_{k = 2..2*p} A122193(p,k)*C(n+2,k+1) for p >= 1. (End) LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened M. Dukes, C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016. FORMULA a(1, p) = 1, a(2, p) = 3^(p-1)-1, a(3, p) = 3^(p-1)*[2^(p-1)-2]+1, ..., a(2*p-3, p) = [ (6*p^4-20*p^3+21*p^2-7*p)*(2*p-4)! ]/[3*2^(p-1)], a(2*p-2, p) = [ (p^2-p)*(2*p-3)! ]/2^(p-2), a(2*p-1, p) = [ (p-1)*(2*p-3)! ]/2^(p-2). a(i, p) = Sum_{k=1..[2*i+1+(-1)^(i-1)]/4} [ C(i-1, 2*k-2)*C(i-2*k+3, i-2*k+1)^(p-1) -C(i-1, 2*k-1)*C(i-2*k+2, i-2*k)^(p-1) ] From Peter Bala, Mar 08 2018: (Start) The following remarks assume row and column indices start at 0. T(n,k) = Sum_{i = 0..k} (-1)^(k-i)*C(k,i)*C(i+2,2)^n. Equivalently, let v_n denote the sequence (1, 3^n, 6^n, 10^n, ...) regarded as an infinite column vector, where 1, 3, 6, 10, ... is the sequence of triangular numbers A000217. Then the n-th row of this table is determined by the matrix product P^(-1)*v_n, where P denotes Pascal's triangle A007318. Cf. A122193. T(n+1,k) = C(k+2,2)*T(n,k) + 2*C(k+1,2)*T(n,k-1) + C(k,2)*T(n,k-2), with boundary conditions T(n,0) = 1 for all n and T(n,k) = 0 for k > 2*n. Let R(n,x) denote the n-th row polynomial. R(n+1,x) = 1/2!*(1 + x)^2*(d/dx)^2 (x^2*R(n,x)). R(n,x) = Sum_{i >= 0} binomial(i+2,2)^n*x^i/(1 + x)^(i+1). R(n,x) = (1 + x)^2 o (1 + x)^2 o ... o (1 + x)^2 (n factors), where o denotes the black diamond product of power series defined in Dukes and White. Note the polynomial x^2 o ... o x^2 (n factors) is the n-th row polynomial of A122193. x^2*R(n,x) = (1 + x)^2 * the n-th row polynomial of A122193 (End) EXAMPLE Row 3 contains 1,8,19,18,6, so Sum_{i=1..n} C(i+1,2)^3 = (n+2) * C(n+1,2) * [ a(1,3)/3 + a(2,3)*C(n-1,1)/4 + a(3,3)*C(n-1,2)/5 + a(4,3)*C(n-1,3)/6 + a(5,3)*C(n-1,4)/7 ] = [ (n+2)*(n+1)*n/2 ] * [ 1/3 + (8/4)*C(n-1,1) + (19/5)*C(n-1,2) + (18/6)*C(n-1,3) + (6/7)*C(n-1,4). Cf. A085438 for more details. From Peter Bala, Mar 08 2018: (Start) Table begins n=0 |1 n=1 |1 2 1 n=2 |1 8 19 18 6 n=3 |1 26 163 432 564 360 90 n=4 |1 80 1135 6354 18078 28800 26100 12600 2520 ... Row 2: C(i+2,2)^2 = C(i,0) + 8*C(i,1) + 19*C(i,2) + 18*C(i,3) + 6*C(i,4). Hence, Sum_{i = 0..n-1} C(i+2,2)^2 = C(n,1) + 8*C(n,2) + 19*C(n,3) + 18*C(n,4) + 6*C(n,5). (End) MAPLE seq(seq(add( (-1)^(k-i)*binomial(k, i)*binomial(i+2, 2)^n, i = 0..k), k = 0..2*n), n = 0..8); # Peter Bala, Mar 08 2018 MATHEMATICA a[i_, p_] := Sum[Binomial[i - 1, 2*k - 2]*Binomial[i - 2*k + 3, i - 2*k + 1]^(p - 1) - Binomial[i - 1, 2*k - 1]*Binomial[i - 2*k + 2, i - 2*k]^(p - 1), {k, 1, (2*i + 1 + (-1)^(i - 1))/4}]; Table[If[p == 1, 1, a[i, p]], {p, 1, 10}, {i, 1, 2*p - 1}]//Flatten (* G. C. Greubel, Nov 23 2017 *) a[i_, p_]:=(-1)^i HypergeometricPFQ[ConstantArray[3, p]~Join~{2-i}, ConstantArray[1, p], 1]; Table[a[i, p], {p, 0, 10}, {i, 2, 2 p+2}]//Flatten (* Jonathan Burns, Mar 20 2018 *) PROG (PARI) {a(i, p) = sum(k=1, (2*i + 1 + (-1)^(i - 1))/4, binomial(i - 1, 2*k - 2)*binomial(i - 2*k + 3, i - 2*k + 1)^(p - 1) - binomial(i - 1, 2*k - 1)*binomial(i - 2*k + 2, i - 2*k)^(p - 1))}; for(p=1, 8, for(i=1, 2*p-1, print1(if(p==1, 1, a(i, p)), ", "))) \\ G. C. Greubel, Nov 23 2017 (GAP) Flat(List([0..6], n->List([0..2*n], k->Sum([0..k], i->(-1)^(k-i)*Binomial(k, i)*Binomial(i+2, 2)^n)))); # Muniru A Asiru, Mar 22 2018 CROSSREFS Cf. A000292, A024166, A024166, A085438, A085439, A085440, A085441, A085442, A087107, A000332, A086020, A086021, A086022, A087108, A000389, A086023, A086024, A087109, A000579, A086025, A086026, A087110, A000580, A086027, A086028, A087111, A027555, A086029, A086030. Cf. A000217, A122193. Sequence in context: A167400 A225848 A165889 * A144946 A332717 A260374 Adjacent sequences: A087124 A087125 A087126 * A087128 A087129 A087130 KEYWORD easy,nonn,tabf AUTHOR André F. Labossière, Aug 11 2003 EXTENSIONS Edited by Dean Hickerson, Aug 16 2003 STATUS approved

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Last modified April 13 17:11 EDT 2024. Contains 371644 sequences. (Running on oeis4.)