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

%I A108125 #34 Sep 08 2022 08:45:19
%S A108125 0,1,2,3,8,20,84,266,1728,5400,62400,105072,3831552,-6751056,
%T A108125 393519840,-2647115280,63975266304,-747746339328,15068270944512,
%U A108125 -236303000537472,4743900735943680,-88955061630656256,1889497707746457600,-40170338917074339840,919697318458416414720
%N A108125 Expansion of e.g.f.: x/(1 - log(1+x)).
%H A108125 Robert Israel, <a href="/A108125/b108125.txt">Table of n, a(n) for n = 0..414</a>
%F A108125 (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
%F A108125 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
%F A108125 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
%p A108125 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
%p A108125 A:= proc(n) option remember;
%p A108125   (-(n^2-2*n)*procname(n-1) + add(binomial(n,j)*procname(j)*procname(n-j),j=1..n-1))/(n-1) end proc:
%p A108125 A(0):= 0: A(1):= 1:
%p A108125 map(A, [$0..50]); # _Robert Israel_, Jul 13 2015
%t A108125 Range[0, 25]! CoefficientList[Series[x/(1 - Log[1 + x]), {x, 0, 25}], x] (* _Vincenzo Librandi_, Jul 06 2015 *)
%o A108125 (Maxima)
%o A108125 a[n]:=if n<2 then n else n!*sum(a[n-k]*(-1)^(k+1)/(k*
%o A108125 (n-k)!),k,1,n-1); makelist(a[n],n,0,100); /* _Tani Akinari_, Nov 25 2017 */
%o A108125 (PARI) x='x+O('x^30); concat([0], Vec(serlaplace(x/(1 - log(1+x))))) \\ _G. C. Greubel_, Aug 30 2018
%o A108125 (Magma) m:=30; R<x>:=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
%K A108125 sign
%O A108125 0,3
%A A108125 _Roger L. Bagula_, Jun 26 2005