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Expansion of e.g.f. Product_{k>=1} 1/(1 + log(1 - x)*x^k).
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%I #13 Jan 07 2019 03:34:20

%S 1,0,2,9,68,490,5184,53928,696352,9545184,147901680,2437886880,

%T 44593856064,861936989472,17988878376000,398199273907680,

%U 9386173867046400,233068382185213440,6117261434418069504,168414066137504272896,4867992707164288773120,147081824197157871866880,4641822165217412602183680

%N Expansion of e.g.f. Product_{k>=1} 1/(1 + log(1 - x)*x^k).

%H Robert Israel, <a href="/A322612/b322612.txt">Table of n, a(n) for n = 0..427</a>

%F E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} log(1/(1 - x))^d/d ) * x^k).

%F a(n) ~ c * n! / r^n, where r = 0.74075364335169502373416717320773551326074821766... is the root of the equation r*log(1-r) = -1 and c = 1 / (r*(r/(1-r) - log(1-r)) * Product_{k>=2} (1 + log(1-r)*r^k) ) = 16.634865259935976898139371781860039862... - _Vaclav Kotesovec_, Dec 20 2018

%p seq(coeff(series(factorial(n)*mul((1+log(1-x)*x^k)^(-1),k=1..n),x,n+1), x, n), n = 0 .. 22); # _Muniru A Asiru_, Dec 21 2018

%t nmax = 22; CoefficientList[Series[Product[1/(1 + Log[1 - x] x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Log[1/(1 - x)]^d/d, {d, Divisors[k]}] x^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

%Y Cf. A227682, A265953, A322613.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Dec 20 2018