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Expansion of e.g.f. 1/(exp(x) + log(1 - x)).
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%I #19 Mar 11 2022 08:33:14

%S 1,0,0,1,5,23,139,1069,9365,90971,981647,11697167,152304591,

%T 2149063421,32668289913,532328418153,9256383832665,171066343532055,

%U 3348245897484091,69189708307509195,1505284330388457451,34391324279752372105,823258887611521993045

%N Expansion of e.g.f. 1/(exp(x) + log(1 - x)).

%H Seiichi Manyama, <a href="/A352146/b352146.txt">Table of n, a(n) for n = 0..444</a>

%F a(0) = 1; a(n) = Sum_{k=1..n} ((k-1)! - 1) * binomial(n,k) * a(n-k).

%F a(n) ~ n! * (1-r) / ((1 - (1-r)*exp(r)) * r^(n+1)), where r = 0.9183335761894542037857295468680123485973875022318007816308... is the root of the equation exp(r) = -log(1-r). - _Vaclav Kotesovec_, Mar 06 2022

%t m = 22; Range[0, m]! * CoefficientList[Series[1/(Exp[x] + Log[1 - x]), {x, 0, m}], x] (* _Amiram Eldar_, Mar 06 2022 *)

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)+log(1-x))))

%o (PARI) a(n) = if(n==0, 1, sum(k=1, n, ((k-1)!-1)*binomial(n, k)*a(n-k)));

%Y Cf. A352138, A352139, A352147.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Mar 06 2022