A377967 Expansion of e.g.f. (1+x)^3 * exp(x*(1+x)^3).

1, 4, 19, 124, 961, 8236, 79339, 840484, 9595009, 117764596, 1542837091, 21406165804, 313381177729, 4822681240924, 77704955681851, 1307128152596116, 22899018541506049, 416756647023727204, 7863586717014612019, 153550319029835965276, 3097694623619639050561

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..508

Formula

a(n) = n! * Sum_{k=0..n} binomial(3*k+3,n-k) / k!.

D-finite with recurrence: -4*(n + 1)*(n + 2)*(n + 3)*(n + 4)*a(n) - 13*(n + 2)*(n + 3)*(n + 4)*a(n + 1) - 15*(n + 4)*(n + 3)*a(n + 2) - 7*a(n + 3)*(n + 4) + a(n + 4)*n + a(n + 5) = 0. - Robert Israel, Feb 24 2026

Maple

f:= gfun:-rectoproc({-4*(n + 1)*(n + 2)*(n + 3)*(n + 4)*a(n) - 13*(n + 2)*(n + 3)*(n + 4)*a(n + 1) - 15*(n + 4)*(n + 3)*a(n + 2) - 7*a(n + 3)*(n + 4) + a(n + 4)*n + a(n + 5),
a(0) = 1, a(1) = 4, a(2) = 19, a(3) = 124, a(4) = 961}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Feb 24 2026

Mathematica

With[{nn=20}, CoefficientList[Series[(1+x)^3 Exp[x*(1+x)^3], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, May 28 2025 *)

Prog

(PARI) a(n, s=3, t=3) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);

Crossrefs

Cf. A361279, A377964, A377966.

Cf. A351767.

Sequence in context: A280939 A121125 A361240 * A099953 A009324 A203236

Adjacent sequences: A377964 A377965 A377966 * A377968 A377969 A377970

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 12 2024

Status

approved