A371140 E.g.f. satisfies A(x) = 1 - x*A(x)^2 * log(1 - x).
1, 0, 2, 3, 56, 270, 5064, 47040, 984416, 14116032, 336538080, 6589416240, 179336461248, 4446985514400, 137520942168960, 4112410749501600, 143445512622458880, 5004065722611594240, 195260931334478223360, 7762385328551718796800, 336051947630616458065920
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Keywords
Programs
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Mathematica
nmax = 20; CoefficientList[Series[(-1 + Sqrt[1 + 4*x*Log[1 - x]])/(2*x*Log[1 - x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 12 2024 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(2/(1+sqrt(1+4*x*log(1-x))))) -
PARI
a(n) = n!*sum(k=0, n\2, (2*k)!/(k+1)!*abs(stirling(n-k, k, 1))/(n-k)!);
Formula
E.g.f.: 2/(1 + sqrt(1+4*x*log(1-x))).
a(n) = n! * Sum_{k=0..floor(n/2)} (2*k)!/(k+1)! * |Stirling1(n-k,k)|/(n-k)!.
a(n) ~ sqrt(2 + 8*r^2/(1-r)) * n^(n-1) / (exp(n) * r^n), where r = 0.436224579489690436773045325306926562580857950193340891933383996... is the root of the equation 4*r*log(1-r) = -1. - Vaclav Kotesovec, Mar 12 2024