A330449 -id:A330449 - cp's OEIS frontend

A330495 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * (k-1)! * sigma_2(k), where sigma_2 = A001157.

1, 6, 37, 307, 2858, 32060, 405830, 5777354, 91400200, 1593023040, 30251766840, 622016655816, 13777150847952, 327040289212320, 8280040187137200, 222696435041359824, 6341359225470493440, 190609840724078576256, 6031297367477133540480, 200389374367707186619776

Offset: 1

Vaclav Kotesovec, Dec 16 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..400

Cf. A330449, A330450, A330493, A330494.

Table[Sum[(-1)^(n-k) * StirlingS1[n, k] * (k-1)! * DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[Log[1/(1 - x)]^k/(k*(1 - Log[1/(1 - x)]^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]

a(n) = sum(k=1, n, (-1)^(n-k)*stirling(n, k, 1)*(k-1)!*sigma(k, 2)); \\ Michel Marcus, Dec 16 2019

E.g.f.: Sum_{k>=1} log(1/(1 - x))^k / (k * (1 - log(1/(1 - x))^k)^2).

a(n) ~ n! * zeta(3) * n * exp(n) / (exp(1) - 1)^(n+2).

A330450 Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)^2).

Original entry on oeis.org

1, 4, 7, 55, -162, 4100, -49030, 779914, -11928008, 198650880, -3538477560, 70414760136, -1571134087824, 38788172175072, -1028732373217200, 28631225505910224, -826097667884640768, 24664145505337921920, -765245501125015575168, 24841409653689047496576

Offset: 1

Ilya Gutkovskiy, Dec 15 2019

sign

Vaclav Kotesovec, Table of n, a(n) for n = 1..400

Cf. A000219, A001157, A008275, A318250, A330354, A330449, A330495.

nmax = 20; CoefficientList[Series[Sum[Log[1 + x]^k/(k (1 - Log[1 + x]^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]

E.g.f.: -Sum_{k>=1} k * log(1 - log(1 + x)^k).
E.g.f.: log(Product_{k>=1} 1 / (1 - log(1 + x)^k)^k).
exp(Sum_{n>=1} a(n) * (exp(x) - 1)^n / n!) = g.f. of A000219.
a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * sigma_2(k), where sigma_2 = A001157.
Conjecture: a(n) ~ n! * (-1)^n * zeta(3) * n * exp(n) / (8 * (exp(1) - 1)^(n+2)). - Vaclav Kotesovec, Dec 16 2019

cp's OEIS Frontend

A330495 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * (k-1)! * sigma_2(k), where sigma_2 = A001157.

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