A052717 Expansion of e.g.f. x*(1 - sqrt(1 - 4*x))/2.

0, 0, 2, 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 673

Crossrefs

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052718, A052719, A052720, A052721, A052722, A052723.

Cf. A000108, A052711.

Programs

Magma

[n le 1 select 0 else Factorial(n)*Catalan(n-2): n in [0..30]]; // G. C. Greubel, May 28 2022

Maple

spec := [S,{C=Union(B,Z),B=Prod(C,C),S=Prod(Z,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
with(combinat):with(combstruct):a[0]:=0:for n from 1 to 30 do a[n]:=sum((count(Permutation(n*2-2),size=n-1)),j=0..n) od: seq(a[n], n=0..22); # Zerinvary Lajos, May 03 2007

Mathematica

With[{nn=20},CoefficientList[Series[x (1-Sqrt[1-4x])/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 20 2015 *)
Table[Boole[n==1] + n!*CatalanNumber[n-2], {n, 0, 30}] (* G. C. Greubel, May 28 2022 *)

MuPAD

combinat::catalan(n)*(n+2)! $ n = 0..15; // Zerinvary Lajos, Feb 15 2007

SageMath

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

Formula

Recurrence: a(1)=0, a(3)=6, a(2)=2, n*a(n+1) = (4*n^2 - 2*n - 6)*a(n).

a(n) = n!*A000108(n-2) = A052711(n), n > 2. - R. J. Mathar, Oct 26 2013

G.f.: x*(d/dx)(x^2 * Hypergeometric2F0([1, 1/2], [], 4*x)). - G. C. Greubel, May 28 2022