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%I #11 Feb 28 2024 10:48:41
%S 0,1,4,21,172,1605,19206,275611,4653160,91082025,2040433930,
%T 51423366951,1440390172860,44391110385661,1491847772350222,
%U 54276959307240195,2124662283059851216,89017250958419937873,3973938501512332468242,188293439203215046203583
%N Expansion of e.g.f. -LambertW(-x) * Product_{k>=1} (1+x^k).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>
%F a(n) ~ c * n^(n-1), where c = QPochhammer(-1, exp(-1))/2 = 1.67792868498935419743907236983684684... - _Vaclav Kotesovec_, Jul 21 2018
%t nmax = 20; CoefficientList[Series[-LambertW[-x]*Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
%t Table[n!*Sum[PartitionsQ[n-k]*k^(k-1)/k!, {k, 1, n}], {n, 0, 20}]
%Y Cf. A000009, A000169, A317103.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Jul 21 2018
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