A308861 Expansion of e.g.f. 1/(1 - x*(1 + x)*exp(x)).

1, 1, 6, 39, 352, 3965, 53556, 844123, 15204960, 308118105, 6937562980, 171826160231, 4642588564032, 135891789038629, 4283619809941668, 144674451274329075, 5211965027738046016, 199498704931954788785, 8085413817213212761668, 345895984008645703002559

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..396

Formula

E.g.f.: 1 / (1 - Sum_{k>=1} k^2*x^k/k!).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).

a(n) ~ n! / (r^(n+1) * exp(r) * (1 + 3*r + r^2)), where r = A201941 = 0.44413022882396659058546632949098466707932096994213775695918... is the root of the equation exp(r)*r*(1 + r) = 1. - Vaclav Kotesovec, Jun 29 2019

Mathematica

nmax = 19; CoefficientList[Series[1/(1 - x (1 + x) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

Prog

(PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1 - x*(1 + x)*exp(x)))) \\ Michel Marcus, Mar 10 2022

Crossrefs

Cf. A000290, A006153, A033453, A033462, A302189, A308862.

Sequence in context: A246571 A031972 A356439 * A124577 A352839 A006678

Adjacent sequences: A308858 A308859 A308860 * A308862 A308863 A308864

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 29 2019

Status

approved