A346847 -id:A346847 - cp's OEIS frontend

A346845 E.g.f.: log(1 + x) / (1 - x)^3.

1, 5, 29, 186, 1374, 11352, 105048, 1070640, 11978640, 145558080, 1914027840, 27035890560, 408891369600, 6585851059200, 112656894336000, 2038285492992000, 38915729475840000, 781515776369664000, 16475855040820224000, 363685261902133248000, 8391522945839007744000

Offset: 1

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000217, A001711, A024167, A109792, A346846, A346847.

nmax = 21; CoefficientList[Series[Log[1 + x]/(1 - x)^3, {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! Sum[(-1)^(k + 1) Binomial[n - k + 2, 2]/k , {k, 1, n}], {n, 1, 21}]
Table[n!*(((-1)^n*(2*n + 5) - 4*n - 5)/8 + (n+1)*(n+2)*(Log[2] - (-1)^n * LerchPhi[-1, 1, 1 + n])/2), {n, 1, 21}] // Simplify (* Vaclav Kotesovec, Aug 06 2021 *)

my(x='x+O('x^25)); Vec(serlaplace(log(1+x)/(1-x)^3)) \\ Michel Marcus, Aug 06 2021

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

a(n) ~ log(2) * n^2 * n! / 2. - Vaclav Kotesovec, Aug 06 2021

A346846 E.g.f.: log(1 + x) / (1 - x)^4.

Original entry on oeis.org

1, 7, 50, 386, 3304, 31176, 323280, 3656880, 44890560, 594463680, 8453128320, 128473430400, 2079045964800, 35692494566400, 648044312832000, 12406994498304000, 249834635947008000, 5278539223415808000, 116768100285720576000, 2699047267616544768000, 65071515565786447872000

Offset: 1

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000292, A001716, A024167, A109792, A346845, A346847.

nmax = 21; CoefficientList[Series[Log[1 + x]/(1 - x)^4, {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! Sum[(-1)^(k + 1) Binomial[n - k + 3, 3]/k , {k, 1, n}], {n, 1, 21}]

my(x='x+O('x^25)); Vec(serlaplace(log(1+x)/(1-x)^4)) \\ Michel Marcus, Aug 06 2021

a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * binomial(n-k+3,3) / k.

a(n) ~ log(2) * n^3 * n! / 6. - Vaclav Kotesovec, Aug 06 2021

cp's OEIS Frontend

A346845 E.g.f.: log(1 + x) / (1 - x)^3.

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