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Expansion of Sum_{k>0} k*(k+1)/2 * x^k / (1 - (-x)^k)^3.
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%I #11 Jul 17 2023 04:34:22

%S 1,0,12,9,30,0,56,60,126,0,132,126,182,0,420,316,306,0,380,330,798,0,

%T 552,888,875,0,1296,630,870,0,992,1536,1914,0,2100,1467,1406,0,2652,

%U 2360,1722,0,1892,1518,4860,0,2256,4872,3234,0,4488,2106,2862,0,5060

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

%H Seiichi Manyama, <a href="/A163920/b163920.txt">Table of n, a(n) for n = 1..10000</a>

%F a(4n+2) = 0.

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

%F a(n) = (n/4) * Sum_{d|n} (-1)^(n+d) * (d+1) * (n/d+1). - _Seiichi Manyama_, Jul 17 2023

%t 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 *)

%o (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))}

%Y Cf. A143520 (variant), A034715.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Aug 06 2009