A052713 - OEIS

%I #22 May 30 2022 02:30:53

%S 0,2,8,96,1920,53760,1935360,85155840,4428103680,265686220800,

%T 18066663014400,1373066389094400,115337576683929600,

%U 10611057054921523200,1061105705492152320000,114599416193152450560000

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

%C Has a square root singularity.

%H G. C. Greubel, <a href="/A052713/b052713.txt">Table of n, a(n) for n = 0..330</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=669">Encyclopedia of Combinatorial Structures 669</a>

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

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

%F a(n+1) = ((2*n)!/n!)*2^(n+1). - _Zerinvary Lajos_, Sep 25 2006

%F a(n) = n!*A025225(n). - _R. J. Mathar_, Oct 18 2013

%F G.f.: (1- 2F0([1,-1/2], [], 8*x))/2. - _R. J. Mathar_, Jan 25 2020

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

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

%o (SageMath) [2^n*factorial(n)*catalan_number(n-1) + bool(n==0)/2 for n in (0..30)] # _G. C. Greubel_, May 29 2022

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

%Y Cf. A000108, A025225.

%K easy,nonn

%O 0,2

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