%I #15 Feb 06 2024 16:20:46

%S 1,-2,4,-6,-8,150,-972,3682,6256,-289746,3300460,-21071622,-27876312,

%T 3156947014,-53217341660,494232431250,175171749088,-113735274256290,

%U 2613309376750812,-32653995355358678,36013529538641560,10227377502146048118,-305630239215263764076

%N Expansion of e.g.f. 1/(1 + 2*x*exp(x)).

%F a(n) = n! * Sum_{k=0..n} (-2)^(n-k) * (n-k)^k/k!.

%F a(0) = 1 and a(n) = -2 * n * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.

%t With[{nn=30},CoefficientList[Series[1/(1+2x Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Feb 06 2024 *)

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+2*x*exp(x))))

%o (PARI) a(n) = n!*sum(k=0, n, (-2)^(n-k)*(n-k)^k/k!);

%o (PARI) a(n) = if(n==0, 1, -2*n*sum(k=0, n-1, binomial(n-1, k)*a(k)));

%Y Column k=2 of A351776.

%Y Cf. A351762.

%K sign

%O 0,2

%A _Seiichi Manyama_, Feb 19 2022