A360770 Expansion of Sum_{k>0} (x * (k + x^k))^k.

1, 5, 27, 260, 3125, 46684, 823543, 16777472, 387420498, 10000003125, 285311670611, 8916100495009, 302875106592253, 11112006826381559, 437893890380860625, 18446744073726328848, 827240261886336764177, 39346408075296925015353

Offset

1,2

Links

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

Formula

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

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

Mathematica

a[n_] := DivisorSum[n, #^(# - n/# + 1) * 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, (x*(k+x^k))^k))
(PARI) a(n) = sumdiv(n, d, d^(d-n/d+1)*binomial(d, n/d-1));

Crossrefs

Cf. A260148, A338693, A360712, A360771.

Cf. A360774, A360775, A360776.

Sequence in context: A230563 A204266 A266204 * A360726 A360712 A300621

Adjacent sequences: A360767 A360768 A360769 * A360771 A360772 A360773

Keyword

nonn

Author

Seiichi Manyama, Feb 20 2023

Status

approved