A360732 Expansion of Sum_{k>0} (k * x * (1 + (k * x)^k))^k.

1, 5, 27, 288, 3125, 48907, 823543, 17039360, 387479538, 10048828125, 285311670611, 8929262337009, 302875106592253, 11116754387067959, 437894195556640625, 18448995890703106048, 827240261886336764177, 39347760450413560593753

Offset: 1

Author

Seiichi Manyama, Feb 18 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 1..385

Crossrefs

Cf. A360712, A360726.

Programs

Mathematica

a[n_] := DivisorSum[n, #^n * Binomial[#, n/# - 1] &]; Array[a, 20] (* Amiram Eldar, Aug 09 2023 *)

PARI

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x*(1+(k*x)^k))^k))

PARI

a(n) = sumdiv(n, d, d^n*binomial(d, n/d-1));

Formula

a(n) = Sum_{d|n} d^n * binomial(d,n/d-1).

If p is an odd prime, a(p) = p^p.