A387969 Expansion of e.g.f. (1+x) * exp(x^2*(1+x)^2).

1, 1, 2, 18, 84, 420, 3720, 31080, 243600, 2434320, 26641440, 283076640, 3293801280, 42404281920, 554325932160, 7519387075200, 109612690387200, 1661012652499200, 25817843674022400, 420212956132569600, 7132591138709836800, 124371851362683724800, 2238662687984500377600

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(2*k+1,n-2*k)/k!.

From Vaclav Kotesovec, Oct 21 2025: (Start)

a(n) = 2*(n-1)*a(n-2) + 2*(n-2)*(3*n-2)*a(n-3) + 4*(n-3)*(n-2)*n*a(n-4).

a(n) ~ n^((3*n+1)/4) * 2^((n-3)/2) * exp(1/64 - 3*n^(1/4)/2^(13/2) - sqrt(n)/16 + n^(3/4)/sqrt(2) - 3*n/4) * (1 + 66551/(40960*sqrt(2)*n^(1/4)) + 6793308403/(20132659200*sqrt(n))). (End)

Mathematica

nmax = 20; CoefficientList[Series[(1+x) * E^(x^2*(1+x)^2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 21 2025 *)
Table[n!*Sum[Binomial[2*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(2*k+1, n-2*k)/k!);
(Magma) [Factorial(n) * &+[Binomial(2*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, A387968, A387970.

Sequence in context: A091940 A068605 A343514 * A360301 A070171 A387922

Adjacent sequences: A387966 A387967 A387968 * A387970 A387971 A387972

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 12 2025

Status

approved