A387970 Expansion of e.g.f. (1+x) * exp(x^2*(1+x)^3).

1, 1, 2, 24, 156, 900, 8400, 96600, 1009680, 11264400, 151532640, 2177461440, 31514978880, 487688160960, 8230011229440, 145043141443200, 2635313083603200, 50358644214278400, 1013552304154790400, 21147853208709427200, 455850291533641804800, 10213553685619841356800

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

From Vaclav Kotesovec, Oct 21 2025: (Start)

a(n) = 2*(n-1)*a(n-2) + (n-2)*(9*n-7)*a(n-3) + (n-3)*(n-2)*(12*n-5)*a(n-4) + 5*(n-4)*(n-3)*(n-2)*n*a(n-5).

a(n) ~ 5^(n/5 - 7/10) * exp(108/15625 - 273*n^(1/5)/(31250*5^(1/5)) - 59*n^(2/5)/(625*5^(2/5)) + 3*n^(3/5)/(25*5^(3/5)) + 3*n^(4/5)/5^(4/5) - 4*n/5) * n^((4*n+1)/5) * (1 + 35198424/(9765625*5^(4/5)*n^(1/5))). (End)

Mathematica

nmax = 20; CoefficientList[Series[(1+x) * E^(x^2*(1+x)^3), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 21 2025 *)
Table[n!*Sum[Binomial[3*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(3*k+1, n-2*k)/k!);
(Magma) [Factorial(n) * &+[Binomial(3*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, A387969.

Cf. A377964.

Sequence in context: A189247 A234352 A241623 * A288443 A108476 A157053

Adjacent sequences: A387967 A387968 A387969 * A387971 A387972 A387973

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 12 2025

Status

approved