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%I #11 Jan 04 2024 19:32:38
%S 1,1,33,8294,8790208,28436662624,228929520628448,3983602580423420352,
%T 135150778123405293748224,8262821715336263175482769408,
%U 855516444430388524429593124012032,142657102263368111456587968163250896896,36753801552552818015956675623665562408714240
%N a(0) = 1; a(n) = (n!)^5 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^5.
%H Alois P. Heinz, <a href="/A336261/b336261.txt">Table of n, a(n) for n = 0..120</a>
%F a(n) = (n!)^5 * [x^n] 1 / (1 - polylog(5,x)).
%F 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
%p b:= proc(n) option remember; `if`(n=0, 1,
%p add(b(n-i)/i^5, i=1..n))
%p end:
%p a:= n-> n!^5*b(n):
%p seq(a(n), n=0..14); # _Alois P. Heinz_, Jan 04 2024
%t 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}]
%t nmax = 12; CoefficientList[Series[1/(1 - PolyLog[5, x]), {x, 0, nmax}], x] Range[0, nmax]!^5
%Y Cf. A007840, A336197, A336258, A336259, A336260.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 15 2020
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