The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #5 Jun 11 2015 06:25:09
%S 1,3,20,158,1307,11352,102538,954904,9112038,88723163,878714118,
%T 8829998320,89848944237,924291213496,9600148608620,100565064076006,
%U 1061498376477423,11281275452880277,120635822090127386,1297256892395670322,14021436433125959714
%N a(n) = [x^n] Product_{k=1..n} 1/(x^k*(1-x^k)^2).
%F a(n) ~ c * d^n / n^(5/2), where d = 12.0708016857156441729965623654557363850943928675996965027830903372727... = r^3/(r-1)^2, where r is the root of the equation polylog(2, 1-r) + (3*log(r)^2)/4 = 0, c = 8.36819319541... .
%t Table[SeriesCoefficient[1/Product[x^k*(1-x^k)^2, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
%t Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}]^2, {x, 0, n*(n+3)/2}], {n, 0, 25}]
%Y Cf. A258788, A258789, A258793, A258794.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Jun 10 2015