A381984 E.g.f. A(x) satisfies A(x) = exp(x) * B(x), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

1, 2, 9, 94, 1649, 40146, 1246057, 47004014, 2087644449, 106709890114, 6170322084041, 398219508589662, 28376096583546769, 2212797385807852754, 187441592012756668329, 17139223549605292448686, 1682551982313514625386817, 176505773149909540258262274, 19704960849698723062181296009

Offset

0,2

Comments

For each positive integer k, the sequence obtained by reducing a(n) modulo k is a periodic sequence with period dividing k. For example, modulo 6 the sequence becomes [1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, ...] with period 6. Cf. A047974. - Peter Bala, Mar 13 2025

Links

Table of n, a(n) for n=0..18.

Formula

a(n) = n! * Sum_{k=0..n} A001764(k)/(n-k)!.

From Peter Bala, Mar 13 2025: (Start)

a(n) = hypergeom([-n, 1/3, 2/3], [3/2], -27/4).

2*(2*n + 1)*a(n) = (27*n^2 - 19*n + 4)*a(n-1) - 2*(n - 1)*(27*n - 25)*a(n-2) + 27*(n - 1)*(n - 2)*a(n-3) with a(0) = 0, a(1) = 2 and a(2) = 9. (End)

a(n) ~ 3^(3*n + 1/2) * n^(n + 1/2) / (2^(2*n + 3/2) * exp(n - 4/27) * n^(3/2)). - Vaclav Kotesovec, Mar 14 2025

Maple

seq(simplify(hypergeom([-n, 1/3, 2/3], [3/2], -27/4)), n = 0..18); # Peter Bala, Mar 13 2025

Mathematica

Table[HypergeometricPFQ[{-n, 1/3, 2/3}, {3/2}, -27/4], {n, 0, 20}] (* Vaclav Kotesovec, Mar 14 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n, binomial(3*k+1, k)/((3*k+1)*(n-k)!));

Crossrefs

Cf. A381985, A381986, A381987.

Cf. A001763, A001764.

Sequence in context: A292467 A291622 A058156 * A011837 A106343 A086992

Adjacent sequences: A381981 A381982 A381983 * A381985 A381986 A381987

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 11 2025

Status

approved