A052681 - OEIS

%I #23 Jun 10 2022 07:34:53

%S 1,0,2,18,48,840,9360,90720,1653120,25764480,442713600,9540115200,

%T 201659673600,4744989849600,123531638630400,3325415917824000,

%U 97123590660096000,3021564701675520000,98526128957448192000

%N Expansion of e.g.f. (1-x)/(1 - x - x^2 - 2*x^3 + 2*x^4).

%H G. C. Greubel, <a href="/A052681/b052681.txt">Table of n, a(n) for n = 0..400</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=629">Encyclopedia of Combinatorial Structures 629</a>

%F E.g.f.: (1-x)/(1 - x - x^2 - 2*x^3 + 2*x^4).

%F Recurrence: a(0)=1, a(1)=0, a(2)=2, a(3)=18, a(n+4) = (n+4)*a(n+3) + (12 + 7*n + n^2)*a(n+2) + (48 + 52*n + 18*n^2 + 2*n^3)*a(n+1) - 2*(n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*a(n).

%F a(n) = (n!/353)*Sum_{alpha=RootOf(1 - Z - z^2 - 2*Z^3 + 2*Z^4)} (18 + 106*alpha - 33*alpha^2 - 28*alpha^3)*alpha^(-1-n).

%F a(n) = n!*A052546(n). - _R. J. Mathar_, Nov 27 2011

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

%t With[{nn=20},CoefficientList[Series[(1-x)/(1-x-x^2-2x^3+2x^4),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, May 23 2014 *)

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1-x)/(1-x-x^2-2*x^3+2*x^4) ))); // _G. C. Greubel_, Jun 09 2022

%o (SageMath)

%o def A052681_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( (1-x)/(1-x-x^2-2*x^3+2*x^4) ).egf_to_ogf().list()

%o A052681_list(40) # _G. C. Greubel_, Jun 09 2022

%Y Cf. A000142, A052546.

%K easy,nonn

%O 0,3

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