A052723 Expansion of e.g.f. (1 - x - sqrt(1-2*x+x^2-4*x^3))/(2*x).

0, 0, 2, 6, 24, 240, 2880, 35280, 524160, 9434880, 188697600, 4151347200, 101548339200, 2727435110400, 79332244992000, 2488504322304000, 83879464660992000, 3021209014247424000, 115754916599562240000

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 679

Crossrefs

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722.

Cf. A023431.

Programs

Maple

spec := [S,{B=Prod(S,S),C=Union(B,S,Z),S=Prod(Z,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(n!*add(binomial(n-2-k,2*k)*binomial(2*k,k)/(k+1), k=0..floor((n-2)/3)), n=0..18);  # Mark van Hoeij, May 12 2013

Mathematica

With[{nn=20},CoefficientList[Series[(1-x-Sqrt[1-2x+x^2-4x^3])/(2x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 19 2017 *)
a[n_]:= a[n]= n!*Sum[Binomial[n-k-2,2*k]*CatalanNumber[k], {k,0,Floor[(n-2)/2]}];
Table[a[n], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

SageMath

def A052723(n): return factorial(n)*sum( binomial(n-k-2, 2*k)*catalan_number(k) for k in (0..(n-2)//2) )
[A052723(n) for n in (0..30)] # G. C. Greubel, May 28 2022

Formula

D-finite with recurrence: a(0) = a(1) = 0, a(2) = 2, a(3) = 6, a(4) = 24, (n+4)*a(n+3) = (15 + 11*n + 2*n^2)*a(n+2) - (6 + 11*n + 6*n^2 + n^3)*a(n+1) - (12 - 2*n - 32*n^2 - 22*n^2 - 4*n^4)*a(n).

a(n) = n!*A023431(n-2). - R. J. Mathar, Oct 18 2013