A336261 a(0) = 1; a(n) = (n!)^5 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^5.

1, 1, 33, 8294, 8790208, 28436662624, 228929520628448, 3983602580423420352, 135150778123405293748224, 8262821715336263175482769408, 855516444430388524429593124012032, 142657102263368111456587968163250896896, 36753801552552818015956675623665562408714240

Offset: 0

Author

Ilya Gutkovskiy, Jul 15 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..120

Crossrefs

Cf. A007840, A336197, A336258, A336259, A336260.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)/i^5, i=1..n))
    end:
a:= n-> n!^5*b(n):
seq(a(n), n=0..14);  # Alois P. Heinz, Jan 04 2024

Mathematica

a[0] = 1; a[n_] := a[n] = (n!)^5 Sum[a[k]/(k! (n - k))^5, {k, 0, n - 1}]; Table[a[n], {n, 0, 12}]
nmax = 12; CoefficientList[Series[1/(1 - PolyLog[5, x]), {x, 0, nmax}], x] Range[0, nmax]!^5

Formula

a(n) = (n!)^5 * [x^n] 1 / (1 - polylog(5,x)).

a(n) ~ (n!)^5 / (polylog(4,r) * r^n), where r = 0.96581751668950729310276791428... is the root of the equation polylog(5,r) = 1. - Vaclav Kotesovec, Jul 15 2020