OFFSET
0,3
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
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 *)
PROG
(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)!);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 12 2024
STATUS
approved