A390105 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x/(1-x)) ).

1, 2, 16, 236, 5168, 151152, 5549248, 245561696, 12729558528, 756900258560, 50793014134784, 3797894975887872, 313143572782422016, 28227755647783000064, 2761905593240611430400, 291523243713345926770688, 33019523633733444546265088, 3994854155132971594500734976

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

E.g.f. A(x) satisfies A(x) = exp(2*x*A(x)/(1-x*A(x))).

E.g.f.: exp(2 * B(x)), where B(x) is the e.g.f. of A376093.

E.g.f.: C(x)^2, where C(x) is the e.g.f. of A361093.

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

a(n) ~ exp(sqrt(3) - 1 - (2 - sqrt(3))*n) * n^(n-1) / (3^(1/4) * (2 - sqrt(3))^n). - Vaclav Kotesovec, Oct 25 2025

Mathematica

Table[2 * n! * Sum[(2*n+2)^(k-1) * Binomial[n-1, n-k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 25 2025 *)

Prog

(PARI) a(n, q=2, r=2, s=2, t=1, u=0) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial((r*u+1)*n+((s-r)*u+t-1)*k+q*u-1, n-k)/k!);
(Magma) [2*Factorial(n) * &+[(2*n+2)^(k-1)*Binomial(n-1, n-k) / Factorial(k) : k in [0..n]] : n in [0..25] ]; // Vincenzo Librandi, Nov 28 2025

Crossrefs

Cf. A052873, A390106.

Cf. A361093, A376093.

Sequence in context: A233288 A181218 A301581 * A376389 A280394 A376392

Adjacent sequences: A390102 A390103 A390104 * A390106 A390107 A390108

Keyword

nonn

Author

Seiichi Manyama, Oct 25 2025

Status

approved