A052715 Expansion of e.g.f. (1-2*x-sqrt(1-4*x))/2 - x*(1-2*x-sqrt(1-4*x)) - x^2.

0, 0, 0, 0, 24, 480, 10080, 241920, 6652800, 207567360, 7264857600, 282291609600, 12067966310400, 563171761152000, 28496491114291200, 1554354060779520000, 90929712555601920000, 5679609738088366080000

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 671

Crossrefs

Cf. A000108, A002057.

Programs

Magma

[n le 3 select 0 else 2*(n-3)*Factorial(n-1)*Catalan(n-2): n in [0..30]]; // G. C. Greubel, May 27 2022

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[(1-2x-Sqrt[1-4x])/2-x(1-2x- Sqrt[ 1-4x])- x^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 18 2016 *)

SageMath

[0]+[4*factorial(n-1)*binomial(2*n-5, n-4) for n in (1..30)] # G. C. Greubel, May 27 2022

Formula

D-finite with recurrence: a(1)=0; a(2)=0; a(3)=0; a(4)=24; 4*(-n-3+2*n^2)*a(n) +2*(-1-3*n)*a(n+1) +a(n+2) =0.

a(n) = n!*A002057(n-4). - R. J. Mathar, Oct 18 2013