A052711 - OEIS

%I #25 May 30 2022 02:24:09

%S 0,0,0,6,48,600,10080,211680,5322240,155675520,5189184000,

%T 194075481600,8045310873600,366061644748800,18134130709094400,

%U 971471287987200000,55956746188062720000,3448334483839365120000

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

%H G. C. Greubel, <a href="/A052711/b052711.txt">Table of n, a(n) for n = 0..350</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=667">Encyclopedia of Combinatorial Structures 667</a>

%F D-finite with recurrence: a(1)=0, a(2)=0, a(3)=6, a(4)=48, n*a(n+1) = 2*(n+1)*(2*n-3)*a(n).

%F From _R. J. Mathar_, Oct 18 2013: (Start)

%F a(n) = n!*A000108(n-2).

%F a(n) = A052717(n), n>2. (End)

%F G.f.: x*(1 - 4*x - 2F0([-1/2,2], [], 4*x))/2. - _R. J. Mathar_, Jan 25 2020

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

%t With[{nn=20},CoefficientList[Series[x (1-2x-Sqrt[1-4x])/2,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 05 2016 *)

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

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

%Y Cf. A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722, A052723.

%Y Cf. A000108, A052717.

%K easy,nonn

%O 0,4

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