A336258 a(0) = 1; a(n) = (n!)^2 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^2.

1, 1, 5, 58, 1208, 39476, 1861372, 119587224, 10040970816, 1067383279872, 140110136642304, 22256626639796352, 4207858001708629248, 933704296260740939520, 240293228328619963492608, 70992050129486593239246336, 23863916105454465092261412864

Offset: 0

Author

Ilya Gutkovskiy, Jul 15 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..247

Crossrefs

Cf. A007840, A074707, A102221, A336243, A336259, A336260, A336261.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)/i^2, i=1..n))
    end:
a:= n-> n!^2*b(n):
seq(a(n), n=0..16);  # Alois P. Heinz, Jan 04 2024

Mathematica

a[0] = 1; a[n_] := a[n] = (n!)^2 Sum[a[k]/(k! (n - k))^2, {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[1/(1 - PolyLog[2, x]), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

a(n) = (n!)^2 * [x^n] 1 / (1 - polylog(2,x)).

a(n) ~ (n!)^2 / (-log(1-r) * r^n), where r = 0.76154294453204558806805187241... is the root of the equation polylog(2,r) = 1. - Vaclav Kotesovec, Jul 15 2020