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A305307 Expansion of e.g.f. 1/(1 - log(1 + x)/(1 - x)).

%I A305307 #10 Aug 08 2021 13:32:50
%S A305307 1,1,3,17,120,1084,11642,146446,2101656,33958344,609431232,
%T A305307 12033015840,259163792016,6047213451408,151953760489008,
%U A305307 4091057804809104,117485988199385088,3584814699783432960,115816462543697120640,3949619921174717629056,141780511159572486530304,5344008726418981985707776
%N A305307 Expansion of e.g.f. 1/(1 - log(1 + x)/(1 - x)).
%C A305307 a(n)/n! is the invert transform of [1, 1 - 1/2, 1 - 1/2 + 1/3, 1 - 1/2 + 1/3 - 1/4, 1 - 1/2 + 1/3 - 1/4 + 1/5, ...].
%H A305307 Alois P. Heinz, <a href="/A305307/b305307.txt">Table of n, a(n) for n = 0..410</a>
%H A305307 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A305307 E.g.f.: 1/(1 - Sum_{k>=1} (A058313(k)/A058312(k))*x^k).
%F A305307 a(n) ~ n! * (2 - LambertW(exp(2))) / ((1 + 1/LambertW(exp(2))) * (LambertW(exp(2)) - 1)^(n+1)). - _Vaclav Kotesovec_, Aug 08 2021
%e A305307 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 120*x^4/4! + 1084*x^5/5! + 11642*x^6/6! + ...
%p A305307 g:= proc(n) g(n):= `if`(n=1, 0, g(n-1))-(-1)^n/n end:
%p A305307 b:= proc(n) option remember; `if`(n=0, 1,
%p A305307       add(g(j)*b(n-j), j=1..n))
%p A305307     end:
%p A305307 a:= n-> b(n)*n!:
%p A305307 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 29 2018
%t A305307 nmax = 21; CoefficientList[Series[1/(1 - Log[1 + x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
%t A305307 nmax = 21; CoefficientList[Series[1/(1 - Sum[Sum[(-1)^(j + 1)/j, {j, 1, k}] x^k , {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
%t A305307 a[0] = 1; a[n_] := a[n] = Sum[((-1)^(k + 1) LerchPhi[-1, 1, k + 1] + Log[2]) a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 21}]
%Y A305307 Cf. A006252, A024167, A058312, A058313, A305306.
%K A305307 nonn
%O A305307 0,3
%A A305307 _Ilya Gutkovskiy_, May 29 2018