A393955 - OEIS

%I #13 Mar 18 2026 04:42:08

%S 1,2,10,108,1840,43000,1279584,46291280,1972707328,96793120800,

%T 5374542400000,333181247151808,22810832109895680,1709414039122395008,

%U 139174640542003978240,12233003976328500000000,1154537363767101233299456,116449585838640863629672960,12500573099908938063034318848

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

%F E.g.f.: 1 + log( (1/x) * Series_Reversion( x*exp(-x*(2+x)) ) ).

%F E.g.f.: B(x)^2, where B(x) is the e.g.f. of A192949.

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

%F a(n) ~ (1 + sqrt(3))^(n + 1/2) * n^(n-1) / (sqrt(2) * 3^(1/4) * exp((1 - sqrt(3)/2)*n)). - _Vaclav Kotesovec_, Mar 18 2026

%t Join[{1}, Table[(n-1)! * Sum[n^k*(2*n)^(n - 2*k)*Binomial[n-k, k]/(n-k)!, {k, 0, Floor[n/2]}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Mar 18 2026 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((1+serreverse(x*exp(-x*(2+x))))^2))

%Y Cf. A192949, A393954.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 04 2026