%I #10 Mar 27 2019 03:51:43
%S 1,1,8,63,628,7405,103266,1630195,28812344,561715353,11971270270,
%T 276322667071,6867229990644,182651988444133,5174629835814362,
%U 155498722020145995,4938797154614179696,165259917542803746097,5809661798192528407542,214032701720169039806551,8244827039453943163648940
%N Expansion of e.g.f. Product_{k>=1} 1/(1 - exp(x)*x^k)^k.
%F E.g.f.: Product_{k>=1} 1/(1 - exp(x)*x^k)^k.
%F a(n) ~ c * n! / LambertW(1)^n, where c = 1/(1 + LambertW(1)) * Product_{j>=1} 1/(1 - LambertW(1)^j)^(j+1) = 115.50749040505570853455997830821388214033876679679... - _Vaclav Kotesovec_, Apr 07 2018
%e Product_{k>=1} 1/(1 - exp(x)*x^k)^k = 1 + x/1! + 8*x^2/2! + 63*x^3/3! + 628*x^4/4! + 7405*x^5/5! + 103266*x^6/6! + ...
%p a:=series(mul(1/(1-exp(x)*x^k)^k,k=1..100),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # _Paolo P. Lava_, Mar 26 2019
%t nmax = 20; CoefficientList[Series[Product[1/(1 - Exp[x] x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%Y Cf. A265952, A265953.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 07 2018