A302399 Expansion of e.g.f. Product_{k>=1} 1/(1 - exp(x)*x^k)^k.

1, 1, 8, 63, 628, 7405, 103266, 1630195, 28812344, 561715353, 11971270270, 276322667071, 6867229990644, 182651988444133, 5174629835814362, 155498722020145995, 4938797154614179696, 165259917542803746097, 5809661798192528407542, 214032701720169039806551, 8244827039453943163648940

Offset: 0

Author

Ilya Gutkovskiy, Apr 07 2018

Keywords

• nonn

Examples

			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! + ...
		

Crossrefs

Cf. A265952, A265953.

Programs

Maple

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

Mathematica

nmax = 20; CoefficientList[Series[Product[1/(1 - Exp[x] x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: Product_{k>=1} 1/(1 - exp(x)*x^k)^k.

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