A308863 Expansion of e.g.f. (1 + LambertW(-x))/(1 + 2*LambertW(-x)).

1, 1, 6, 57, 736, 11985, 235296, 5403937, 142073856, 4206560769, 138483596800, 5017244970441, 198363105460224, 8498001799768273, 392127481640165376, 19388814120804416625, 1022681739669784231936, 57317273018414456262273, 3401527253966521309200384

Offset: 0

Author

Ilya Gutkovskiy, Jun 29 2019

Keywords

• nonn

Crossrefs

Cf. A000312, A002866, A086331, A218688, A277610, A308861, A308862.

Programs

Mathematica

nmax = 18; CoefficientList[Series[(1 + LambertW[-x])/(1 + 2 LambertW[-x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

Formula

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

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^k * a(n-k).

a(n) ~ sqrt(Pi) * 2^(n - 3/2) * n^(n + 1/2) / exp(n/2). - Vaclav Kotesovec, Jun 29 2019