A327249 -id:A327249 - cp's OEIS frontend

A360712 Expansion of Sum_{k>0} (k * x * (1 + k*x^k))^k.

1, 5, 27, 272, 3125, 46915, 823543, 16781312, 387421218, 10000078125, 285311670611, 8916102153177, 302875106592253, 11112006865911623, 437893890381640625, 18446744074783358976, 827240261886336764177, 39346408075327943829273

Offset: 1

Seiichi Manyama, Feb 17 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A217668, A327249, A338693, A359700, A360618.

a[n_] := DivisorSum[n, #^(#+n/#-1) * Binomial[#, n/# - 1] &]; Array[a, 20] (* Amiram Eldar, Aug 09 2023 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x*(1+k*x^k))^k))

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

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

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

A360733 Expansion of Sum_{k>0} (x * (1 + (k * x)^k))^k.

Original entry on oeis.org

1, 2, 1, 9, 1, 98, 1, 1025, 2188, 15626, 1, 692836, 1, 5764802, 97656251, 201326593, 1, 36138519442, 1, 409470748547, 14242684529830, 3138428376722, 1, 10019491686645761, 476837158203126, 3937376385699290, 5403406870691968357, 19704673338472752470, 1

Offset: 1

Seiichi Manyama, Feb 18 2023

nonn

Winston de Greef, Table of n, a(n) for n = 1..599

Cf. A327249, A360732.

a[n_] := DivisorSum[n, #^(n-#) * Binomial[#, n/# - 1] &]; Array[a, 30] (* Amiram Eldar, Aug 09 2023 *)

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

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

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

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

A360756 Expansion of Sum_{k>0} (x * (1 + 2 * x^k))^k.

Original entry on oeis.org

1, 3, 1, 5, 1, 11, 1, 9, 13, 11, 1, 45, 1, 15, 41, 49, 1, 79, 1, 117, 85, 23, 1, 297, 81, 27, 145, 309, 1, 483, 1, 481, 221, 35, 561, 1165, 1, 39, 313, 2121, 1, 1143, 1, 1365, 2437, 47, 1, 4081, 449, 3411, 545, 2341, 1, 4699, 5281, 4889, 685, 59, 1, 20445, 1, 63, 6217

Offset: 1

Seiichi Manyama, Feb 19 2023

nonn

Cf. A327249, A360755.

a[n_] := DivisorSum[n, 2^(n/# - 1) * Binomial[#, n/# - 1] &]; Array[a, 60] (* Amiram Eldar, Aug 02 2023 *)

my(N=70, 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-1));

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

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

cp's OEIS Frontend

A360712 Expansion of Sum_{k>0} (k * x * (1 + k*x^k))^k.

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A360733 Expansion of Sum_{k>0} (x * (1 + (k * x)^k))^k.

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A360756 Expansion of Sum_{k>0} (x * (1 + 2 * x^k))^k.

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