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A305306 Expansion of e.g.f. 1/(1 + log(1 - x)/(1 - x)).

%I A305306 #22 May 10 2023 08:23:35
%S A305306 1,1,5,35,324,3744,51902,839362,15513096,322550616,7451677632,
%T A305306 189366303840,5249764639248,157666361452560,5099445234111888,
%U A305306 176713626295062384,6531995374500741888,256537368987293878272,10667901271715707803264,468261481657502075856768,21635865693957558515860224
%N A305306 Expansion of e.g.f. 1/(1 + log(1 - x)/(1 - x)).
%C A305306 a(n)/n! is the invert transform of [1, 1 + 1/2, 1 + 1/2 + 1/3, 1 + 1/2 + 1/3 + 1/4, ...].
%H A305306 Alois P. Heinz, <a href="/A305306/b305306.txt">Table of n, a(n) for n = 0..395</a>
%H A305306 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A305306 E.g.f.: 1/(1 - Sum_{k>=1} (A001008(k)/A002805(k))*x^k).
%F A305306 a(n) ~ n! / ((1/LambertW(1)^2 - 1) * (1 - LambertW(1))^n). - _Vaclav Kotesovec_, Aug 08 2021
%F A305306 a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A006153(k). - _Seiichi Manyama_, May 10 2023
%e A305306 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 35*x^3/3! + 324*x^4/4! + 3744*x^5/5! + 51902*x^6/6! + ...
%p A305306 H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
%p A305306 b:= proc(n) option remember; `if`(n=0, 1,
%p A305306       add(H(j)*b(n-j), j=1..n))
%p A305306     end:
%p A305306 a:= n-> b(n)*n!:
%p A305306 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 29 2018
%t A305306 nmax = 20; CoefficientList[Series[1/(1 + Log[1 - x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
%t A305306 nmax = 20; CoefficientList[Series[1/(1 - Sum[HarmonicNumber[k] x^k , {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
%t A305306 a[0] = 1; a[n_] := a[n] = Sum[HarmonicNumber[k] a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 20}]
%o A305306 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)/(1-x)))) \\ _Seiichi Manyama_, May 10 2023
%Y A305306 Cf. A000254, A001008, A002805, A007840, A128044, A128045, A305307.
%Y A305306 Cf. A006153, A087761.
%K A305306 nonn
%O A305306 0,3
%A A305306 _Ilya Gutkovskiy_, May 29 2018