A052691 Expansion of e.g.f. (1-x)/(1-2*x+x^2-x^3).

1, 1, 2, 12, 96, 840, 8640, 105840, 1491840, 23587200, 413683200, 7983360000, 168129561600, 3835844812800, 94239732787200, 2480658276096000, 69651967537152000, 2077925954936832000, 65637135231123456000

Offset

0,3

Links

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 639

Formula

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

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

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

a(n) = n!*A005251(n+1). - R. J. Mathar, Nov 27 2011

Maple

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

Mathematica

With[{nn=20}, CoefficientList[Series[(1-x)/(1-2x+x^2-x^3), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, May 09 2018 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( (1-x)/(1-2*x+x^2-x^3) ))); // G. C. Greubel, Jun 02 2022
(SageMath) [factorial(n)*sum(binomial(n-j, 2*j) for j in (0..n//3)) for n in (0..40)] # G. C. Greubel, Jun 02 2022

Crossrefs

Cf. A000142, A005251.

Sequence in context: A239837 A239838 A306258 * A371040 A292419 A322543

Adjacent sequences: A052688 A052689 A052690 * A052692 A052693 A052694

Keyword

easy,nonn

Author

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

Status

approved