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

Author

Ilya Gutkovskiy, Aug 05 2021

Keywords

• nonn

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Formula

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