A360771 Expansion of Sum_{k>=0} (x * (2 + x^k))^k.

1, 2, 5, 8, 20, 32, 77, 128, 288, 518, 1104, 2048, 4313, 8192, 16832, 32848, 66568, 131072, 264688, 524288, 1053737, 2097824, 4205568, 8388608, 16803744, 33554442, 67162112, 134222336, 268550704, 536870912, 1073999165, 2147483648, 4295493376, 8589962752

Offset: 0

Author

Seiichi Manyama, Feb 20 2023

Keywords

• nonn

Crossrefs

Cf. A217668, A260148.

Programs

Mathematica

a[n_] := DivisorSum[n, 2^(# - n/# + 1) * Binomial[#, n/# - 1] &]; a[0] = 1; Array[a, 30, 0] (* Amiram Eldar, Aug 02 2023 *)

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (x*(2+x^k))^k))

PARI

a(n) = if(n==0, 1, sumdiv(n, d, 2^(d-n/d+1)*binomial(d, n/d-1)));

Formula

a(n) = Sum_{d|n} 2^(d-n/d+1) * binomial(d,n/d-1) for n > 0.

If p is an odd prime, a(p) = 2^p.