A052719 Expansion of e.g.f. (1 - 2*x*sqrt(1-4*x))*(1 - sqrt(1-4*x))/4.

0, 0, 0, 6, 72, 1080, 20160, 453600, 11975040, 363242880, 12454041600, 476367091200, 20113277184000, 929233405900800, 46630621823385600, 2525825348766720000, 146886458743664640000, 9127944221927731200000

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 675

Crossrefs

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052720, A052721, A052722, A052723.

Cf. A000108, A000245.

Programs

Maple

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

Mathematica

Table[If[n<2, 0, 3*(n-2)*(n-1)!*CatalanNumber[n-2]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

SageMath

[0,0]+[3*(n-2)*factorial(n-1)*catalan_number(n-2) for n in (2..30)] # G. C. Greubel, May 28 2022

Formula

D-finite with recurrence: a(1)=0, a(2)=0, a(3)=6, a(n+2) = (2 + 5*n)*a(n+1) + (6 + 2*n - 4*n^2)*a(n)

a(n) = n!*A000245(n-2). - R. J. Mathar, Oct 26 2013

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

G.f.: 6*x^3*Hypergeometric2F0([2, 3/2], [], 4*x).

E.g.f.: (1/4)*(1 + 2*x - 8*x^2 - (1 + 2*x)*sqrt(1-4*x)). (End)