A052688 Expansion of e.g.f. x/((1-x)*(1-x^3)).

0, 1, 2, 6, 48, 240, 1440, 15120, 120960, 1088640, 14515200, 159667200, 1916006400, 31135104000, 435891456000, 6538371840000, 125536739328000, 2134124568576000, 38414242234368000, 851515702861824000

Offset

0,3

Links

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 636

Formula

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

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

a(n) = n!*A002264(n+2). - R. J. Mathar, Nov 27 2011

Maple

spec := [S, {S=Prod(Z, Sequence(Z), Sequence(Prod(Z, Z, Z)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
G(x):=x/(1-x)/(1-x^3): f[0]:=G(x): for n from 1 to 19 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..19); # Zerinvary Lajos, Apr 03 2009

Mathematica

With[{nn=20}, CoefficientList[Series[x/((1-x)(1-x^3)), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Dec 24 2018 *)
Table[n!*Floor[(n+2)/3], {n, 0, 40}] (* G. C. Greubel, Jun 02 2022 *)

Prog

(SageMath) [factorial(n)*((n+2)//3) for n in (0..40)] # G. C. Greubel, Jun 02 2022

Crossrefs

Cf. A000142, A002264.

Sequence in context: A098710 A337907 A052614 * A052657 A230714 A092143

Adjacent sequences: A052685 A052686 A052687 * A052689 A052690 A052691

Keyword

easy,nonn

Author

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

Status

approved