A052711 Expansion of e.g.f. x*(1 - 2*x - sqrt(1-4*x))/2.

0, 0, 0, 6, 48, 600, 10080, 211680, 5322240, 155675520, 5189184000, 194075481600, 8045310873600, 366061644748800, 18134130709094400, 971471287987200000, 55956746188062720000, 3448334483839365120000

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 667

Crossrefs

Cf. A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722, A052723.

Cf. A000108, A052717.

Programs

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[x (1-2x-Sqrt[1-4x])/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 05 2016 *)
Table[n!*CatalanNumber[n-2] +Boole[n==1] -2*Boole[n==2], {n,0,30}] (* G. C. Greubel, May 30 2022 *)

SageMath

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

Formula

D-finite with recurrence: a(1)=0, a(2)=0, a(3)=6, a(4)=48, n*a(n+1) = 2*(n+1)*(2*n-3)*a(n).

From R. J. Mathar, Oct 18 2013: (Start)

a(n) = n!*A000108(n-2).

a(n) = A052717(n), n>2. (End)

G.f.: x*(1 - 4*x - 2F0([-1/2,2], [], 4*x))/2. - R. J. Mathar, Jan 25 2020