A360752 Expansion of Sum_{k>0} (x * (1 + (2 * x)^k))^k.

1, 3, 1, 9, 1, 41, 1, 65, 193, 161, 1, 2433, 1, 897, 10241, 18433, 1, 66049, 1, 403457, 344065, 22529, 1, 7127041, 5242881, 106497, 9437185, 73629697, 1, 332890113, 1, 940572673, 230686721, 2228225, 9395240961, 18828754945, 1, 9961473, 5234491393, 429517701121, 1

Offset: 1

Author

Seiichi Manyama, Feb 19 2023

Keywords

• nonn

Crossrefs

Cf. A360733.

Programs

Mathematica

a[n_] := DivisorSum[n, 2^(n-#) * Binomial[#, n/# - 1] &]; Array[a, 40] (* Amiram Eldar, Aug 02 2023 *)

PARI

my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, (x*(1+(2*x)^k))^k))

PARI

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

Formula

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

If p is an odd prime, a(p) = 1.