A336259 a(0) = 1; a(n) = (n!)^3 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^3.

1, 1, 9, 278, 20464, 2948824, 735078968, 291153023664, 172201253334528, 145044581320046592, 167609226267379703808, 257816558769660828601344, 514890814087717253133447168, 1307445058678686737908660752384, 4146656933568759002389401276616704

Offset: 0

Author

Ilya Gutkovskiy, Jul 15 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..180

Crossrefs

Cf. A007840, A193436, A336195, A336258, A336260, A336261.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)/i^3, i=1..n))
    end:
a:= n-> n!^3*b(n):
seq(a(n), n=0..14);  # Alois P. Heinz, Jan 04 2024

Mathematica

a[0] = 1; a[n_] := a[n] = (n!)^3 Sum[a[k]/(k! (n - k))^3, {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]
nmax = 14; CoefficientList[Series[1/(1 - PolyLog[3, x]), {x, 0, nmax}], x] Range[0, nmax]!^3

Formula

a(n) = (n!)^3 * [x^n] 1 / (1 - polylog(3,x)).

a(n) ~ (n!)^3 / (polylog(2,r) * r^n), where r = 0.86512013798076629268795131756... is the root of the equation polylog(3,r) = 1. - Vaclav Kotesovec, Jul 15 2020