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%I #15 Feb 28 2024 10:48:37
%S 0,1,4,27,220,2265,27246,393421,6548536,126257697,2767122010,
%T 68387691141,1882488882660,57198150690577,1900138953826582,
%U 68502961685976525,2662089147552365296,110887849449189768513,4926985461324765096498,232544882903837769171829
%N Expansion of e.g.f. -LambertW(-x) * Product_{k>=1} 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 = 1/QPochhammer(exp(-1)) = 1.98244090741287370368568246556131... - _Vaclav Kotesovec_, Jul 21 2018
%t nmax = 20; CoefficientList[Series[-LambertW[-x]*Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
%t Table[n!*Sum[PartitionsP[n-k]*k^(k-1)/k!, {k, 1, n}], {n, 0, 20}]
%Y Cf. A000041, A000169, A252761, A291332, A291333, A317104.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Jul 21 2018