A052721 Expansion of e.g.f. x*(1-2*x)*(1 - 2*x - sqrt(1-4*x))/2 - x^3.

0, 0, 0, 0, 0, 120, 2880, 70560, 1935360, 59875200, 2075673600, 79913433600, 3387499315200, 156883562035200, 7884404656128000, 427447366714368000, 24869664972472320000, 1545805113445232640000, 102232975285590589440000

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 677

Crossrefs

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052722, A052723.

Cf. A000108, A002057.

Programs

Maple

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

Mathematica

Table[If[n<5, 0, 2*n*(n-2)!*(n-4)*CatalanNumber[n-3]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

SageMath

def A052721(n):
    if (n<5): return 0
    else: return 2*n*factorial(n-2)*(n-4)*catalan_number(n-3)
[A052721(n) for n in (0..30)] # G. C. Greubel, May 28 2022

Formula

D-finite with recurrence: a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=120, a(6)=2880, (n+2)*a(n+2) = (6*n^2 + 8*n - 8)*a(n+1) + (40 + 44*n = 4*n^2 - 8*n^3)*a(n).

a(n) = 2*Pi^(-1/2)*4^(n-3)*Gamma(n-5/2)*n*(n-4) for n>3. - Mark van Hoeij, Oct 30 2011

a(n) = n!*A002057(n-5). - R. J. Mathar, Oct 18 2013

From G. C. Greubel, May 28 2022: (Start)

G.f.: 4!*x*(d/dx)( x^5 * Hypergeometric2F0([2, 5/2], [], 4*x) ).

E.g.f.: (x/2)*(1 - 4*x + 2*x^2 - (1-2*x)*sqrt(1-4*x)). (End)