A052722 Expansion of e.g.f. (1 - 2*x - sqrt(1-4*x))^2 * (1 - sqrt(1-4*x))/8.

0, 0, 0, 0, 0, 120, 3600, 100800, 3024000, 99792000, 3632428800, 145297152000, 6351561216000, 301699157760000, 15487223431680000, 854894733428736000, 50516506975334400000, 3182539939446067200000, 212985365178313728000000

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 678

Crossrefs

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052723.

Cf. A000108, A000344.

Programs

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[((1-2x-Sqrt[1-4x])^2 (1-Sqrt[1-4x]))/8,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 30 2021 *)
Table[If[n<5, 0, 10*(n-2)!*Binomial[n-3,2]*CatalanNumber[n-3]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

SageMath

def A052722(n):
    if (n<5): return 0
    else: return 10*factorial(n-2)*binomial(n-3,2)*catalan_number(n-3)
[A052722(n) for n in (0..30)] # G. C. Greubel, May 28 2022

Formula

D-finite with recurrence: a(0) = a(1) = a(2) = a(3) = a(4) = 0, a(5)=120, a(n+3) = (9+7*n)*a(n+2) + (14 - 19*n - 13*n^2)*a(n+1) - (20 + 22*n - 2*n^2 - 4*n^3)*a(n).

a(n) = n!*A000344(n-3). - R. J. Mathar, Oct 18 2013

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

G.f.: 5!*x^5*hypergeometric2F0([5/2, 3], [], 4*x).

E.g.f.: (1/2)*(1 - 5*x + 5*x^2 - (1 - 3*x + x^2)*sqrt(1-4*x)). (End)