A052668 Expansion of e.g.f. 1/(1 - 3*x - x^3).

1, 3, 18, 168, 2088, 32400, 603360, 13109040, 325503360, 9092684160, 282219033600, 9635476435200, 358879494758400, 14480588157235200, 629228583138355200, 29295027261916416000, 1454816084780298240000

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..375

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 616

Formula

E.g.f.: 1/(1-3*x-x^3).

a(n) = 3*n*a(n-1) + n*(n-1)*(n-2)*a(n-3), a(0)=1, a(1)=3, a(2)=18.

a(n) = (n!/15) * Sum_{alpha=RootOf(-1+3*_Z+_Z^3)} (4 + alpha + 2*alpha^2) * alpha^(-1-n).

a(n) = n!*A052541(n). - R. J. Mathar, Nov 27 2011

Maple

spec := [S, {S=Sequence(Union(Z, Z, Z, Prod(Z, Z, Z)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

Mathematica

a[n_]:= a[n]= If[n<3, 3^n*n!, 3*n*a[n-1] + n*(n-1)*(n-2)*a[n-3]];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Sep 03 2022 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( 1/(1-3*x-x^3) ))); // G. C. Greubel, Sep 03 2022
(SageMath)
def A052668_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( 1/(1-3*x-x^3) ).egf_to_ogf().list()
A052668_list(40) # G. C. Greubel, Sep 03 2022

Crossrefs

Cf. A052541.

Sequence in context: A319938 A053513 A138211 * A224788 A121423 A074932

Adjacent sequences: A052665 A052666 A052667 * A052669 A052670 A052671

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Status

approved