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

1,2

Links

Table of n, a(n) for n=1..18.

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.

Mathematica

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

Prog

(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));

Crossrefs

Cf. A360732.

Sequence in context: A351737 A047778 A290802 * A367886 A351735 A322233

Adjacent sequences: A360751 A360752 A360753 * A360755 A360756 A360757

Keyword

nonn

Author

Seiichi Manyama, Feb 19 2023

Status

approved