A295239 Expansion of e.g.f. 2/(1 + sqrt(1 + 4*x*exp(x))).

1, -1, 2, -9, 68, -705, 9234, -146209, 2717000, -57986433, 1397949830, -37576332321, 1114326129564, -36141571087297, 1272713716466906, -48360394499269665, 1972269941821097744, -85929979225787811585, 3983422470176606823054, -195765982110500512129057

Offset: 0

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

• sign

Crossrefs

Cf. A000108, A006531, A052895, A295238.

Programs

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}]

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

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