A387968 Expansion of e.g.f. (1+x) * exp(x^2*(1+x)).

1, 1, 2, 12, 36, 180, 1200, 5880, 42000, 317520, 2147040, 18627840, 160332480, 1357836480, 13457283840, 131026896000, 1308193286400, 14405693702400, 156901205260800, 1777343260492800, 21421310643532800, 257805952961356800, 3232211688779673600

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Formula

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

From Vaclav Kotesovec, Oct 21 2025: (Start)

Recurrence: n*a(n) = a(n-1) + (n-1)*(2*n-1)*a(n-2) + 3*(n-2)*(n-1)*(n+1)*a(n-3).

a(n) ~ 3^(n/3 - 5/6) * n^(2*n/3 + 1/3) * exp(4/81 - 2*3^(-7/3)*n^(1/3) + 3^(-2/3)*n^(2/3) - 2*n/3) * (1 + 1924*3^(1/3)/(2187*n^(1/3))). (End)

Mathematica

nmax = 25; CoefficientList[Series[(1+x) * E^(x^2*(1+x)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 21 2025 *)
Table[n!*Sum[Binomial[k+1, n-2*k]/k!, {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Oct 27 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, binomial(k+1, n-2*k)/k!);
(Magma) [Factorial(n) * &+[Binomial(k+1, n-2*k) / Factorial(k) : k in [0..Floor(n/2)]] : n in [0..25] ]; // Vincenzo Librandi, Oct 27 2025

Crossrefs

Cf. A081125, A387969, A387970.

Cf. A018191.

Sequence in context: A169630 A192385 A352281 * A361570 A294464 A366618

Adjacent sequences: A387965 A387966 A387967 * A387969 A387970 A387971

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 12 2025

Status

approved