OFFSET
0,3
FORMULA
E.g.f.: 1/(1 + x*exp(x)/(1 + x*exp(x)/(1 + x*exp(x)/(1 + x*exp(x)/(1 + ...))))), a continued fraction.
a(n) ~ sqrt(2*(1+LambertW(-1/4))) * n^(n-1) / (exp(n) * (LambertW(-1/4))^n). - Vaclav Kotesovec, Nov 18 2017
a(n) = n! * Sum_{k=0..n} (-1)^k * k^(n-k) * A000108(k)/(n-k)!. - Seiichi Manyama, Oct 30 2024
MAPLE
a:=series(2/(1+sqrt(1+4*x*exp(x))), x=0, 20): seq(n!*coeff(a, x, n), n=0..19); # Paolo P. Lava, Mar 27 2019
MATHEMATICA
nmax = 19; CoefficientList[Series[2/(1 + Sqrt[1 + 4 x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[x Exp[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n, k] k! Sum[(-1)^m (m + 1)^(k - m - 1) Binomial[2 m, m]/(k - m)!, {m, 0, k}], {k, 0, n}], {n, 0, 19}]
PROG
(PARI) a(n) = n!*sum(k=0, n, (-1)^k*k^(n-k)*binomial(2*k, k)/((k+1)*(n-k)!)); \\ Seiichi Manyama, Oct 30 2024
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Nov 18 2017
STATUS
approved