A052712 Expansion of e.g.f. (1 + 4*x - sqrt(1-8*x))/8.

0, 1, 2, 24, 480, 13440, 483840, 21288960, 1107025920, 66421555200, 4516665753600, 343266597273600, 28834394170982400, 2652764263730380800, 265276426373038080000, 28649854048288112640000

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Also the number of random walk labelings of the 2 X (n-1) king's graph, for n > 1. - Sela Fried, Apr 14 2023

Links

• G. C. Greubel, Table of n, a(n) for n = 0..330
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 668
• Sela Fried and Toufik Mansour, Graph labelings obtainable by random walks, arXiv:2304.05728 [math.CO], 2023.

Crossrefs

Cf. A052711, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722, A052723.

Cf. A000108, A003645.

Programs

Maple

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

Mathematica

Table[n!*2^(n-2)*CatalanNumber[n-1] +Boole[n==1]/2 +Boole[n==0]/4, {n,0,30}] (* G. C. Greubel, May 30 2022 *)

SageMath

[2^(n-2)*factorial(n)*catalan_number(n-1) +bool(n==0)/8 +bool(n==1)/2 for n in (0..30)] # G. C. Greubel, May 30 2022

Formula

D-finite with recurrence: a(0) = 0, a(1)=1, a(2)=2, a(n+1) = 4*(2*n-1)*a(n).

a(n) = 8^(n+1)*Gamma(n+3/2)/sqrt(Pi).

a(n) = n!*A003645(n-2), n>1. - R. J. Mathar, Oct 18 2013

G.f.: (1 + 4*x - 2F0([1,-1/2], [], 8*x))/8. - R. J. Mathar, Jan 25 2020