A052692 Expansion of e.g.f. (1-x^4)/(1-x-x^4).

1, 1, 2, 6, 24, 240, 2160, 20160, 201600, 2540160, 36288000, 558835200, 9101030400, 161902540800, 3138418483200, 65383718400000, 1443672502272000, 33790305669120000, 838710955450368000

Offset

0,3

Links

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 640

Formula

E.g.f.: (1-x^4)/(1-x-x^4).

Recurrence: a(0)=1, a(1)=1, a(2)=2, a(3)=6, a(4)=24, a(n) = n*a(n-1) + n*(n-1)*(n-2)*(n-3)*a(n-4).

a(n) = (n!/283)*Sum_{alpha=RootOf(-1 +z +Z^4)} (36 - 9*alpha + 64*alpha^2 + 48*alpha^3)*alpha^(-1-n).

a(n) = n!*A003269(n), n>0. - R. J. Mathar, Nov 27 2011

Maple

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

Mathematica

b[n_]:= b[n]= If[n<4, 1-Boole[n==0], b[n-1]+b[n-4]]; (* b = A003269 *)
a[n_]:= n!*b[n] +Boole[n==0];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jun 01 2022 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( (1-x^4)/(1-x-x^4) ))); // G. C. Greubel, Jun 01 2022
(SageMath)
@CachedFunction
def A003269(n):
    if (n<4): return 1-bool(n==0)
    else: return A003269(n-1) + A003269(n-4)
def A052692(n): return factorial(n)*A003269(n) +bool(n==0)
[A052692(n) for n in (0..40)] # G. C. Greubel, Jun 01 2022

Crossrefs

Cf. A000142, A003269.

Sequence in context: A346121 A052597 A052632 * A052723 A114778 A292751

Adjacent sequences: A052689 A052690 A052691 * A052693 A052694 A052695

Keyword

easy,nonn

Author

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

Status

approved