A362300 a(n) = n! * Sum_{k=0..floor(n/3)} (n/3)^k * binomial(n-2*k,k)/(n-2*k)!.

1, 1, 1, 7, 33, 101, 1681, 14211, 72577, 1906633, 23242401, 166218911, 5966236321, 95016917997, 873707885233, 39767572858651, 781865428682241, 8787169718273681, 484500265577706817, 11335266937098816183, 150554918241183405601, 9749671976020428623221

Offset

0,4

Comments

Let k be a positive integer. It appears that reducing this sequence modulo k produces an eventually periodic sequence with period a multiple of k. For example, modulo 9 the sequence becomes [1, 1, 1, 7, 6, 2, 7, 0, 1, 1, 0, 8, 7, 0, 7, 7, 0, 8, 1, 0, 1, 7, 0, 2, 7, 0, 1, 1, 0, 8, 7, 0, 7, 7, 0, 8, 1, 0, 1, 7, 0, 2, 7, 0, 1, ...], with an apparent period [2, 7, 0, 1, 1, 0, 8, 7, 0, 7, 7, 0, 8, 1, 0, 1, 7, 0] of length 18 starting at a(5). - Peter Bala, Apr 16 2023

Links

Winston de Greef, Table of n, a(n) for n = 0..425

Eric Weisstein's World of Mathematics, Lambert W-Function.

Formula

a(n) = A362043(n,2*n).

a(n) = n! * [x^n] exp(x + n*x^3/3).

E.g.f.: exp( ( -LambertW(-x^3) )^(1/3) ) / (1 + LambertW(-x^3)).

a(n) ~ (1 + 2*cos(2*Pi*mod(n,3)/3 - sqrt(3)/2)/exp(3/2)) * n^n / (sqrt(3) * exp(2*n/3 - 1)). - Vaclav Kotesovec, Apr 18 2023

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-lambertw(-x^3))^(1/3))/(1+lambertw(-x^3))))

Crossrefs

Cf. A362043, A362293.

Cf. A277614, A362314, A362319.

Sequence in context: A168574 A212106 A370214 * A131211 A213131 A100855

Adjacent sequences: A362297 A362298 A362299 * A362301 A362302 A362303

Keyword

nonn

Author

Seiichi Manyama, Apr 15 2023

Status

approved