A052682 Expansion of e.g.f. (1-x)/(1-x-3*x^2).

1, 0, 6, 18, 288, 2520, 41040, 604800, 11733120, 236234880, 5530291200, 138790713600, 3855483878400, 115075344384000, 3716149018982400, 128239702246656000, 4727462529613824000, 185010460036706304000

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..350
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 630

Crossrefs

Cf. A000142, A052533.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( (1-x)/(1-x-3*x^2) ))); // G. C. Greubel, Jun 04 2022

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[(1-x)/(1-x-3*x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 15 2020 *)
a[n_]:= a[n]= If[n<2, 1-n, n*a[n-1] +3*n*(n-1)*a[n-2]];
Table[a[n], {n,0,40}] (* G. C. Greubel, Jun 04 2022 *)

SageMath

[factorial(n)*sum(binomial(n-k-1, n-2*k)*3^k for k in (0..n//2)) for n in (0..40)] # G. C. Greubel, Jun 04 2022

Formula

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

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

a(n) = (n!/13)*Sum_{alpha=RootOf(-1+Z+3*Z^2)} (-1 + 7*alpha)*alpha^(-1-n).

a(n) = n!*A052533(n). - R. J. Mathar, Nov 27 2011