A109792 Expansion of e.g.f. log(1+x)/(1-x)^2.

1, 3, 14, 70, 444, 3108, 25584, 230256, 2342880, 25771680, 312888960, 4067556480, 57424792320, 861371884800, 13869128448000, 235775183616000, 4264876094976000, 81032645804544000, 1627055289796608000, 34168161085728768000, 754132445894209536000, 17345046255566819328000

Offset: 1

Author

Vladeta Jovovic, Aug 14 2005

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..400

Crossrefs

Cf. A001705, A092692.

Programs

Mathematica

CoefficientList[Series[Log[1+x]/(1-x)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)
a[n_] := n! ((-1)^n (n + 1) LerchPhi[-1, 1, n + 2] + Log[2] (n + 1) + ((-1)^(n + 1) - 1) /2); Table[Simplify[a[n]], {n, 1, 22}] (* Peter Luschny, Jun 22 2022 *)

PARI

for(n=1,25, print1(n!*sum(k=1,n, sum(i=1, k, (-1)^(i+1)/i)), ", ")) \\ G. C. Greubel, Jan 21 2017

Formula

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

a(n) ~ n!*n*log(2). - Vaclav Kotesovec, Jun 27 2013

a(n) = n!*((-1)^n*(n + 1)*LerchPhi(-1, 1, n + 2) + log(2)*(n + 1) + ((-1)^(n + 1) - 1) / 2). - Peter Luschny, Jun 22 2022