%I #14 May 04 2022 08:28:27

%S 1,1,5,32,274,2939,37833,568210,9753280,188342949,4041170695,

%T 95380234366,2455830637412,68501591450447,2057726452045145,

%U 66227424015265178,2273614433910697920,82932491842062712873,3202994529476330549163,130577628147690206429038,5603479009890212632226756

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

%H Seiichi Manyama, <a href="/A305407/b305407.txt">Table of n, a(n) for n = 0..399</a>

%H <a href="/index/Lo#logarithmic">Index entries for sequences related to logarithmic numbers</a>

%F a(n) ~ n! / ((1 + exp(r)/r) * (1 - exp(-r))^(n+1)), where r = 0.62747017959751658496114808922921433658821962606026068561095... is the root of the equation r*exp(1 - exp(-r)) = 1. - _Vaclav Kotesovec_, Mar 26 2019

%F a(0) = 1; a(n) = Sum_{k=1..n} A002104(k) * binomial(n,k) * a(n-k). - _Seiichi Manyama_, May 04 2022

%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 32*x^3/3! + 274*x^4/4! + 2939*x^5/5! + 37833*x^6/6! + ...

%p a:=series(1/(1+log(1-x)*exp(x)),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # _Paolo P. Lava_, Mar 26 2019

%t nmax = 20; CoefficientList[Series[1/(1 + Log[1 - x] Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = Sum[HypergeometricPFQ[{1, 1, 1 - k}, {2}, -1] a[n - k]/(k - 1)!, {k, 1, n}]; Table[n! a[n], {n, 0, 20}]

%Y Cf. A002104, A006153, A007840, A009324, A191365.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 31 2018