A330044 Expansion of e.g.f. exp(x) / (1 - x^3).

1, 1, 1, 7, 25, 61, 841, 5251, 20497, 423865, 3780721, 20292031, 559501801, 6487717237, 44317795705, 1527439916731, 21798729916321, 180816606476401, 7478345832314977, 126737815733490295, 1236785588298582841, 59677199741873516461, 1171057417377450325801

Offset: 0

Author

Ilya Gutkovskiy, Nov 28 2019

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A000522, A087208, A100732, A158757, A330045.

Programs

Magma

[n le 3 select 1 else 1 + 6*Binomial(n-1,3)*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 05 2021

Mathematica

nmax = 22; CoefficientList[Series[Exp[x]/(1 - x^3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[n!/(n - 3 k)!, {k, 0, Floor[n/3]}], {n, 0, 22}]

Sage

[sum(factorial(3*k)*binomial(n, 3*k) for k in (0..n//3)) for n in (0..40)] # G. C. Greubel, Dec 05 2021

Formula

G.f.: Sum_{k>=0} (3*k)! * x^(3*k) / (1 - x)^(3*k + 1).

a(0) = a(1) = a(2) = 1; a(n) = n * (n - 1) * (n - 2) * a(n - 3) + 1.

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

a(n) ~ n! * (exp(1)/3 + 2*cos(sqrt(3)/2 - 2*Pi*n/3) / (3*exp(1/2))). - Vaclav Kotesovec, Apr 18 2020

a(n) = A158757(n, 2*n). - G. C. Greubel, Dec 05 2021