A373773 Expansion of e.g.f. exp(x^3 / (6 * (1 - x)^2)) / (1 - x).

1, 1, 2, 7, 36, 240, 1930, 17990, 189840, 2233000, 28949200, 410009600, 6297999400, 104275571400, 1851050401200, 35065930299400, 705993054166400, 15051593241484800, 338705933426660800, 8021585392026606400, 199416162740963168000, 5191567315003621552000

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-k,n-3*k)/(6^k * k!).

From Vaclav Kotesovec, Jun 18 2024: (Start)

Recurrence: 6*a(n) = 6*(3*n-2)*a(n-1) - 6*(n-1)*(3*n-4)*a(n-2) + 3*(n-2)*(n-1)*(2*n-3)*a(n-3) - (n-3)*(n-2)*(n-1)*a(n-4).

a(n) ~ 3^(-1/3) * exp(19/72 - 3^(-2/3)*n^(1/3) + 3^(2/3)*n^(2/3)/2 - n) * n^(n + 1/6). (End)

Prog

(PARI) a(n) = n!*sum(k=0, n\3, binomial(n-k, n-3*k)/(6^k*k!));

Crossrefs

Cf. A130906, A361597, A373772.

Cf. A361596, A361598.

Cf. A373757.

Sequence in context: A353166 A308750 A088715 * A088313 A201197 A185754

Adjacent sequences: A373770 A373771 A373772 * A373774 A373775 A373776

Keyword

nonn

Author

Seiichi Manyama, Jun 18 2024

Status

approved