A052713 Expansion of e.g.f. (1-sqrt(1-8*x))/2.

0, 2, 8, 96, 1920, 53760, 1935360, 85155840, 4428103680, 265686220800, 18066663014400, 1373066389094400, 115337576683929600, 10611057054921523200, 1061105705492152320000, 114599416193152450560000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

Has a square root singularity.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..330
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 669

Crossrefs

Cf. A052711, A052712, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722, A052723.

Cf. A000108, A025225.

Programs

Maple

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

Mathematica

Table[n!*2^n*CatalanNumber[n-1] + Boole[n==0], {n,0,30}] (* G. C. Greubel, May 29 2022 *)

SageMath

[2^n*factorial(n)*catalan_number(n-1) + bool(n==0)/2 for n in (0..30)] # G. C. Greubel, May 29 2022

Formula

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

a(n+1) = 1/4*8^(n+1)*Gamma(n+1/2)/Pi^(1/2)

a(n+1) = ((2*n)!/n!)*2^(n+1). - Zerinvary Lajos, Sep 25 2006

a(n) = n!*A025225(n). - R. J. Mathar, Oct 18 2013

G.f.: (1- 2F0([1,-1/2], [], 8*x))/2. - R. J. Mathar, Jan 25 2020