A163920 Expansion of Sum_{k>0} k*(k+1)/2 * x^k / (1 - (-x)^k)^3.

1, 0, 12, 9, 30, 0, 56, 60, 126, 0, 132, 126, 182, 0, 420, 316, 306, 0, 380, 330, 798, 0, 552, 888, 875, 0, 1296, 630, 870, 0, 992, 1536, 1914, 0, 2100, 1467, 1406, 0, 2652, 2360, 1722, 0, 1892, 1518, 4860, 0, 2256, 4872, 3234, 0, 4488, 2106, 2862, 0, 5060

Offset: 1

Author

Paul D. Hanna, Aug 06 2009

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A143520 (variant), A034715.

Programs

Mathematica

CoefficientList[Series[Sum[((k(k+1))/2 x^k)/(1-(-x)^k)^3,{k,100}],{x,0,100}],x] (* Harvey P. Dale, May 08 2021 *)

PARI

{a(n) = if( n<1, 0, polcoeff( sum(k=1, n, k*(k+1)/2 * x^k / (1 - (-x)^k)^3, x*O(x^n)), n))}

Formula

a(4n+2) = 0.

a(2n+1) = A034715(2n+1), where A034715 is the Dirichlet convolution of triangular numbers with themselves.

a(n) = (n/4) * Sum_{d|n} (-1)^(n+d) * (d+1) * (n/d+1). - Seiichi Manyama, Jul 17 2023