A360797 Expansion of Sum_{k>0} x^k / (1 - 2 * x^k)^(k+1).

1, 5, 13, 39, 81, 225, 449, 1115, 2345, 5373, 11265, 25483, 53249, 116497, 246405, 529195, 1114113, 2372741, 4980737, 10515511, 22025617, 46204953, 96468993, 201506607, 419432417, 872787997, 1811981789, 3758970975, 7784628225, 16108217801, 33285996545, 68723976779

Offset: 1

Author

Seiichi Manyama, Feb 21 2023

Keywords

• nonn

Crossrefs

Cf. A360798.

Programs

Mathematica

a[n_] := DivisorSum[n, 2^(n/# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 30] (* Amiram Eldar, Jul 31 2023 *)

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-2*x^k)^(k+1)))

PARI

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

Formula

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

If p is prime, a(p) = 1 + p * 2^(p-1).