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%I #14 Nov 07 2017 03:18:00
%S 1,1,6,18,55,150,424,1113,2923,7401,18510,45271,109297,259447,608428,
%T 1407958,3222132,7292198,16340830,36265672,79775931,173999194,
%U 376497975,808471181,1723592762,3649271887,7675809680,16043777217,33332888108,68853608216,141438908854,288994878713,587458691042
%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*k-1)/2).
%C Euler transform of the pentagonal numbers (A000326).
%H Vaclav Kotesovec, <a href="/A278768/b278768.txt">Table of n, a(n) for n = 0..5000</a>
%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(3*k-1)/2).
%F a(n) ~ exp(-Zeta'(-1)/2 - 3*Zeta(3)/(8*Pi^2) - 25*Zeta(3)^3/(6*Pi^8) - 5^(5/4)*Zeta(3)^2/(2^(7/4)*Pi^5) * n^(1/4) - sqrt(5/2)*Zeta(3)/Pi^2 * sqrt(n) + 2^(7/4)*Pi/(3*5^(1/4)) * n^(3/4)) / (2^(155/96) * 5^(11/96) * Pi^(1/24) * n^(59/96)). - _Vaclav Kotesovec_, Dec 02 2016
%p with(numtheory):
%p a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p d^2*(3*d-1)/2, d=divisors(j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Dec 02 2016
%t nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (3 k - 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A000294, A000326, A000335, A023871.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 28 2016
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