A317103 Expansion of e.g.f. -LambertW(-x) * Product_{k>=1} 1/(1-x^k).

0, 1, 4, 27, 220, 2265, 27246, 393421, 6548536, 126257697, 2767122010, 68387691141, 1882488882660, 57198150690577, 1900138953826582, 68502961685976525, 2662089147552365296, 110887849449189768513, 4926985461324765096498, 232544882903837769171829

Offset: 0

Author

Vaclav Kotesovec, Jul 21 2018

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function
• Wikipedia, Lambert W function

Crossrefs

Cf. A000041, A000169, A252761, A291332, A291333, A317104.

Programs

Mathematica

nmax = 20; CoefficientList[Series[-LambertW[-x]*Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
Table[n!*Sum[PartitionsP[n-k]*k^(k-1)/k!, {k, 1, n}], {n, 0, 20}]

Formula

a(n) ~ c * n^(n-1), where c = 1/QPochhammer(exp(-1)) = 1.98244090741287370368568246556131... - Vaclav Kotesovec, Jul 21 2018