A322612 Expansion of e.g.f. Product_{k>=1} 1/(1 + log(1 - x)*x^k).

1, 0, 2, 9, 68, 490, 5184, 53928, 696352, 9545184, 147901680, 2437886880, 44593856064, 861936989472, 17988878376000, 398199273907680, 9386173867046400, 233068382185213440, 6117261434418069504, 168414066137504272896, 4867992707164288773120, 147081824197157871866880, 4641822165217412602183680

Offset: 0

Author

Ilya Gutkovskiy, Dec 20 2018

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..427

Crossrefs

Cf. A227682, A265953, A322613.

Programs

Maple

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

Mathematica

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

Formula

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

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