A317104 Expansion of e.g.f. -LambertW(-x) * Product_{k>=1} (1+x^k).

0, 1, 4, 21, 172, 1605, 19206, 275611, 4653160, 91082025, 2040433930, 51423366951, 1440390172860, 44391110385661, 1491847772350222, 54276959307240195, 2124662283059851216, 89017250958419937873, 3973938501512332468242, 188293439203215046203583

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. A000009, A000169, A317103.

Programs

Mathematica

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

Formula

a(n) ~ c * n^(n-1), where c = QPochhammer(-1, exp(-1))/2 = 1.67792868498935419743907236983684684... - Vaclav Kotesovec, Jul 21 2018