A360754 Expansion of Sum_{k>0} (k * x * (1 + (2 * x)^k))^k.

1, 6, 27, 288, 3125, 47368, 823543, 16793600, 387425673, 10000500000, 285311670611, 8916118771200, 302875106592253, 11112007563452544, 437893890412859375, 18446744108073484288, 827240261886336764177, 39346408077084637733376

Offset: 1

Author

Seiichi Manyama, Feb 19 2023

Keywords

• nonn

Crossrefs

Cf. A360732.

Programs

Mathematica

a[n_] := DivisorSum[n, 2^(n-#) * #^# * Binomial[#, n/# - 1] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)

PARI

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x*(1+(2*x)^k))^k))

PARI

a(n) = sumdiv(n, d, 2^(n-d)*d^d*binomial(d, n/d-1));

Formula

a(n) = Sum_{d|n} 2^(n-d) * d^d * binomial(d,n/d-1).

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