A052679 Expansion of e.g.f. (1-x^2)/(1-x^2-x^3).

1, 0, 0, 6, 0, 120, 720, 5040, 80640, 725760, 10886400, 159667200, 2395008000, 43589145600, 784604620800, 15692092416000, 334764638208000, 7469435990016000, 179266463760384000, 4500868715126784000

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..350

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 627

Formula

E.g.f.: (1 - x^2)/(1 - x^2 - x^3).

Recurrence: a(0)=1, a(1)=0, a(2)=0, a(n+3) = (6 + 5*n + n^2)*a(n+1) + (6 + 11*n + 6*n^2 + n^3)*a(n).

a(n) = (n!/23)*Sum_{alpha=RootOf(Z^3 +Z^2 -1)} (-6 + 9*alpha + 4*alpha^2)*alpha^(-1-n).

a(n) = n!*A000931(n). - R. J. Mathar, Nov 27 2011

Maple

spec := [S, {S=Sequence(Prod(Z, Z, Z, Sequence(Prod(Z, Z))))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
restart: G(x):=(1-x-x^2)/(-1+x^2+x^3): f[0]:=G(x): for n from 1 to 30 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/n, n=1..20); # Zerinvary Lajos, Mar 27 2009

Mathematica

With[{nn=20}, CoefficientList[Series[(1-x^2)/(1-x^2-x^3), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, May 21 2018 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( (1 - x^2)/(1-x^2-x^3) ))); // G. C. Greubel, Jun 11 2022
(SageMath)
@CachedFunction
def A000931(n):
    if (n<3): return bool(n==0)
    else: return A000931(n-2) + A000931(n-3)
def A052679(n): return factorial(n)*A000931(n)
[A052679(n) for n in (0..50)] # G. C. Greubel, Jun 11 2022

Crossrefs

Cf. A000142, A000931.

Sequence in context: A005212 A167028 A246137 * A266218 A240818 A134680

Adjacent sequences: A052676 A052677 A052678 * A052680 A052681 A052682

Keyword

easy,nonn

Author

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

Status

approved