A393955 Expansion of e.g.f. ( 1 + Series_Reversion( x*exp(-x*(2+x)) ) )^2.

1, 2, 10, 108, 1840, 43000, 1279584, 46291280, 1972707328, 96793120800, 5374542400000, 333181247151808, 22810832109895680, 1709414039122395008, 139174640542003978240, 12233003976328500000000, 1154537363767101233299456, 116449585838640863629672960, 12500573099908938063034318848

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

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

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

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)!.

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A192949, A393954.

Sequence in context: A355210 A185396 A003222 * A262145 A355209 A343307

Adjacent sequences: A393952 A393953 A393954 * A393956 A393957 A393958

Keyword

nonn

Author

Seiichi Manyama, Mar 04 2026

Status

approved