A052692 - OEIS

%I #19 Jun 08 2022 03:27:31

%S 1,1,2,6,24,240,2160,20160,201600,2540160,36288000,558835200,

%T 9101030400,161902540800,3138418483200,65383718400000,

%U 1443672502272000,33790305669120000,838710955450368000

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

%H G. C. Greubel, <a href="/A052692/b052692.txt">Table of n, a(n) for n = 0..375</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=640">Encyclopedia of Combinatorial Structures 640</a>

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

%F 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).

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

%F a(n) = n!*A003269(n), n>0. - _R. J. Mathar_, Nov 27 2011

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

%t b[n_]:= b[n]= If[n<4, 1-Boole[n==0], b[n-1]+b[n-4]]; (* b = A003269 *)

%t a[n_]:= n!*b[n] +Boole[n==0];

%t Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Jun 01 2022 *)

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( (1-x^4)/(1-x-x^4) ))); // _G. C. Greubel_, Jun 01 2022

%o (SageMath)

%o @CachedFunction

%o def A003269(n):

%o     if (n<4): return 1-bool(n==0)

%o     else: return A003269(n-1) + A003269(n-4)

%o def A052692(n): return factorial(n)*A003269(n) +bool(n==0)

%o [A052692(n) for n in (0..40)] # _G. C. Greubel_, Jun 01 2022

%Y Cf. A000142, A003269.

%K easy,nonn

%O 0,3

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