A360798 Expansion of Sum_{k>0} x^k / (1 - (2 * x)^k)^(k+1).

1, 5, 13, 45, 81, 321, 449, 1745, 2945, 9153, 11265, 60609, 53249, 230401, 410625, 1259777, 1114113, 7263233, 4980737, 31337473, 44630017, 115367937, 96468993, 937283585, 551550977, 2399256577, 4594597889, 14579646465, 7784628225, 89894944769, 33285996545

Offset: 1

Author

Seiichi Manyama, Feb 21 2023

Keywords

• nonn

Crossrefs

Cf. A360797.

Programs

Mathematica

a[n_] := DivisorSum[n, 2^(n-#) * Binomial[# + n/# - 1, #] &]; Array[a, 30] (* Amiram Eldar, Aug 02 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)*binomial(d+n/d-1, d));

Formula

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

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