A108125 Expansion of e.g.f.: x/(1 - log(1+x)).

0, 1, 2, 3, 8, 20, 84, 266, 1728, 5400, 62400, 105072, 3831552, -6751056, 393519840, -2647115280, 63975266304, -747746339328, 15068270944512, -236303000537472, 4743900735943680, -88955061630656256, 1889497707746457600, -40170338917074339840, 919697318458416414720

Offset: 0

Author

Roger L. Bagula, Jun 26 2005

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..414

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(x/(1 - Log(1+x)))); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // G. C. Greubel, Aug 30 2018

Maple

G:=x/(1-ln(1+x)): Gser:=series(G,x=0,28): 0,seq(n!*coeff(Gser,x^n),n=1..25); # yields the signed sequence
A:= proc(n) option remember;
  (-(n^2-2*n)*procname(n-1) + add(binomial(n,j)*procname(j)*procname(n-j),j=1..n-1))/(n-1) end proc:
A(0):= 0: A(1):= 1:
map(A, [$0..50]); # Robert Israel, Jul 13 2015

Mathematica

Range[0, 25]! CoefficientList[Series[x/(1 - Log[1 + x]), {x, 0, 25}], x] (* Vincenzo Librandi, Jul 06 2015 *)

Maxima

a[n]:=if n<2 then n else n!*sum(a[n-k]*(-1)^(k+1)/(k*
(n-k)!),k,1,n-1); makelist(a[n],n,0,100); /* Tani Akinari, Nov 25 2017 */

PARI

x='x+O('x^30); concat([0], Vec(serlaplace(x/(1 - log(1+x))))) \\ G. C. Greubel, Aug 30 2018

Formula

(n-1) a(n) + (n^2-2n) a(n-1) = Sum_{j=1..n-1} C(n,j) a(j) a(n-j). - Robert Israel, Jul 13 2015

For n > 1, a(n) = n!*Sum_{k=1..n-1} a(n-k)*(-1)^(k+1)/(k*(n-k)!). - Tani Akinari, Nov 25 2017

a(n) ~ n! * (-1)^n / (n * (log(n))^2) * (1 - 2*(gamma + 1) / log(n) + (3*gamma^2 + 6*gamma + 3 - Pi^2/2) / (log(n))^2), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Nov 29 2017