A258348 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)).

1, 0, 2, 6, 15, 32, 79, 172, 397, 860, 1879, 3986, 8462, 17586, 36408, 74366, 150875, 303006, 604511, 1195872, 2350614, 4587484, 8898857, 17154278, 32883109, 62679852, 118858190, 224238730, 421021209, 786793776, 1463796383, 2711552690, 5002097398, 9190449808

Offset: 0

Author

Vaclav Kotesovec, May 27 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000294, A023871, A258344, A258347, A258349, A258350, A258351, A258352.

Programs

Mathematica

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)),{k,1,nmax}],{x,0,nmax}],x]
Clear[a]; a[n_]:= a[n] = 1/n*Sum[(DivisorSigma[3, k]-DivisorSigma[2, k])*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 100}] (* Vaclav Kotesovec, Apr 11 2016, following a suggestion of George Beck *)

Formula

a(n) ~ 1 / (2^(3/2) * 15^(5/48) * Pi^(1/12) * n^(29/48)) * exp(-Zeta'(-1) - Zeta(3)/(4*Pi^2) - 75*Zeta(3)^3 / Pi^8 - 15^(5/4) * Zeta(3)^2 / (2*Pi^5) * n^(1/4) - sqrt(15) * Zeta(3) / Pi^2 * sqrt(n) + 4*Pi / (3*15^(1/4)) * n^(3/4)), where Zeta(3) = A002117, Zeta'(-1) = A084448 = 1/12 - log(A074962).

G.f.: exp(Sum_{k>=1} (sigma_3(k) - sigma_2(k))*x^k/k). - Ilya Gutkovskiy, Aug 22 2018