A324158 -id:A324158 - cp's OEIS frontend

A360810 Expansion of Sum_{k>=0} ( x / (1 - k * x^2) )^k.

1, 1, 1, 2, 5, 11, 29, 81, 229, 696, 2181, 7045, 23653, 81433, 288173, 1046814, 3887749, 14768783, 57275541, 226462801, 912443397, 3741515804, 15603500797, 66134448329, 284660214181, 1243605590897, 5511058189989, 24760003963802, 112726590916645

Offset: 0

Seiichi Manyama, Feb 21 2023

nonn

Cf. A324158, A360782.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k*x^2))^k))

a(n) = sum(k=0, n\2, (n-2*k)^k*binomial(n-k-1, k));

a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n-k-1,k).

A360801 Expansion of Sum_{k>0} (x / (1 - 2 * x^k))^k.

Original entry on oeis.org

1, 3, 5, 13, 17, 51, 65, 169, 281, 603, 1025, 2373, 4097, 8655, 16685, 33969, 65537, 134151, 262145, 530269, 1050481, 2108439, 4194305, 8420201, 16778337, 33607707, 67120565, 134338493, 268435457, 537151131, 1073741825, 2148024289, 4295035145, 8591048739

Offset: 1

Seiichi Manyama, Feb 21 2023

nonn

Cf. A157019, A217670, A324158.

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

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

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

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

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

cp's OEIS Frontend

A360810 Expansion of Sum_{k>=0} ( x / (1 - k * x^2) )^k.

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A360801 Expansion of Sum_{k>0} (x / (1 - 2 * x^k))^k.

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