A360794 -id:A360794 - cp's OEIS frontend

A360795 Expansion of Sum_{k>0} x^k / (1 - (k * x)^k)^(k+1).

1, 3, 4, 17, 6, 211, 8, 1929, 7300, 22601, 12, 1724809, 14, 6703047, 223678576, 738787345, 18, 65630598229, 20, 2119646503661, 24448573943662, 3423809253371, 24, 21453113652593665, 12016296386718776, 4240253019018225, 8255251542208471048, 67251293544533119589, 30

Offset: 1

Seiichi Manyama, Feb 21 2023

nonn

Cf. A360787, A360788.

Cf. A339481, A360794.

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

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

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

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

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

A360823 Expansion of Sum_{k>0} k * x^k / (1 - k * x^k)^(k+1).

Original entry on oeis.org

1, 4, 6, 20, 10, 96, 14, 256, 288, 650, 22, 4200, 26, 4004, 11160, 18784, 34, 70758, 38, 164140, 196098, 136664, 46, 1756728, 393800, 747890, 3287844, 5452076, 58, 22563060, 62, 31220032, 50767926, 20059286, 41640130, 391194396, 74, 99622016, 725647728, 1298396440

Offset: 1

Seiichi Manyama, Feb 22 2023

nonn

Cf. A338658, A360794, A360824.

a[n_] := DivisorSum[n, #^(n/#) * Binomial[# + n/# - 1, #] &]; Array[a, 40] (* Amiram Eldar, Jul 31 2023 *)

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

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

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

If p is prime, a(p) = 2 * p.

A360948 a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(d+n/d-1,d).

Original entry on oeis.org

1, 3, 4, 11, 6, 41, 8, 89, 100, 182, 12, 1088, 14, 723, 2592, 3697, 18, 11804, 20, 29289, 30382, 13037, 24, 246912, 78776, 58554, 374248, 687929, 30, 2567895, 32, 3431585, 4640462, 1182284, 6265548, 37037563, 38, 5246529, 55878240, 128618380, 42, 266983306, 44

Offset: 1

Seiichi Manyama, Feb 26 2023

nonn

Cf. A338662, A360794.

a[n_] := DivisorSum[n, (n/#)^(#-1) * Binomial[# + n/# - 1, #] &]; Array[a, 50] (* Amiram Eldar, Jul 31 2023 *)

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

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

G.f.: Sum_{k>0} (1/k) * (1/(1 - k * x^k)^k - 1).

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

cp's OEIS Frontend

A360795 Expansion of Sum_{k>0} x^k / (1 - (k * x)^k)^(k+1).

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A360823 Expansion of Sum_{k>0} k * x^k / (1 - k * x^k)^(k+1).

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Formula

A360948 a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(d+n/d-1,d).

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