A346890 Expansion of e.g.f. 1 / (1 - x^4 * exp(x) / 4!).

1, 0, 0, 0, 1, 5, 15, 35, 140, 1386, 12810, 92730, 589545, 4234945, 41832791, 483334215, 5401798220, 57262207380, 626438655900, 7740130412796, 107197808258745, 1546730804858085, 22360919412385015, 329241486278715395, 5121840342205301946

Offset

0,6

Links

Seiichi Manyama, Table of n, a(n) for n = 0..482

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * binomial(k,4) * a(n-k).

a(n) ~ n! / ((1 + LambertW(3^(1/4)/2^(5/4))) * 4^(n + 1) * LambertW(3^(1/4)/2^(5/4))^n). - Vaclav Kotesovec, Aug 08 2021

a(n) = n! * Sum_{k=0..floor(n/4)} k^(n-4*k)/(24^k * (n-4*k)!). - Seiichi Manyama, May 13 2022

Mathematica

nmax = 24; CoefficientList[Series[1/(1 - x^4 Exp[x]/4!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 4] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

Prog

(PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^4*exp(x)/4!))) \\ Michel Marcus, Aug 06 2021
(PARI) a(n) = n!*sum(k=0, n\4, k^(n-4*k)/(24^k*(n-4*k)!)); \\ Seiichi Manyama, May 13 2022

Crossrefs

Column k=4 of A351703.

Cf. A006153, A346888, A346889, A346893.

Cf. A000332, A145454.

Sequence in context: A091875 A056413 A032276 * A333932 A065780 A220480

Adjacent sequences: A346887 A346888 A346889 * A346891 A346892 A346893

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 06 2021

Status

approved