A379885 - OEIS

%I #10 Jan 05 2025 09:35:45

%S 1,2,11,118,1885,40266,1080679,34979134,1326825497,57744176914,

%T 2836795756771,155305155441030,9376803979425205,619006372481008474,

%U 44357422104298022399,3429215554499681260366,284496868838293052890033,25212167721275946619910178,2377021703587467346833760315

%N E.g.f. A(x) satisfies A(x) = 1/(exp(-x) - x*A(x)).

%F E.g.f.: 2/(exp(-x) + sqrt(exp(-2*x) - 4*x)).

%F a(n) = n! * Sum_{k=0..n} (2*n-2*k+1)^(k-1) * binomial(2*n-2*k+1,n-k)/k!.

%F a(n) ~ sqrt(1 + LambertW(1/2)) * 2^n * n^(n-1) / (LambertW(1/2)^(n + 1/2) * exp(n)). - _Vaclav Kotesovec_, Jan 05 2025

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(2/(exp(-x)+sqrt(exp(-2*x)-4*x))))

%o (PARI) a(n) = n!*sum(k=0, n, (2*n-2*k+1)^(k-1)*binomial(2*n-2*k+1, n-k)/k!);

%Y Cf. A377890, A377892.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 05 2025