OFFSET
0,2
FORMULA
E.g.f. A(x) satisfies A(x) = exp(2*x*A(x)) / ( 1 - x*exp(x*A(x)) ).
a(n) = n! * Sum_{k=0..n} (n+k+2)^k * binomial(n,k)/(k+1)!.
a(n) ~ sqrt(s/(2 + s*(2*s-3))) * (1-s)^(n + 3/2) * n^(n-1) * s^(2*n+2) / (exp(n) * (1-2*s)^(2*n+2)), where s = 0.438081146547076751360133261776891656173572778544... is the root of the equation s^2 = (1 - 2*s)*exp(s). - Vaclav Kotesovec, Feb 01 2026
MATHEMATICA
nterms=18; CoefficientList[(1/x)*InverseSeries[Series[x*Exp[-2*x]/(1 + x*Exp[-x]), {x, 0, nterms}], x], x]*Range[0, nterms-1]! (* Stefano Spezia, Nov 11 2025 *)
PROG
(PARI) a(n) = n!*sum(k=0, n, (n+k+2)^k*binomial(n, k)/(k+1)!);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Feb 04 2025
STATUS
approved