cp's OEIS Frontend

A052719 Expansion of e.g.f. (1 - 2*x*sqrt(1-4*x))*(1 - sqrt(1-4*x))/4.

%I A052719 #21 May 29 2022 03:15:24
%S A052719 0,0,0,6,72,1080,20160,453600,11975040,363242880,12454041600,
%T A052719 476367091200,20113277184000,929233405900800,46630621823385600,
%U A052719 2525825348766720000,146886458743664640000,9127944221927731200000
%N A052719 Expansion of e.g.f. (1 - 2*x*sqrt(1-4*x))*(1 - sqrt(1-4*x))/4.
%H A052719 G. C. Greubel, <a href="/A052719/b052719.txt">Table of n, a(n) for n = 0..350</a>
%H A052719 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=675">Encyclopedia of Combinatorial Structures 675</a>
%F A052719 D-finite with recurrence: a(1)=0, a(2)=0, a(3)=6, a(n+2) = (2 + 5*n)*a(n+1) + (6 + 2*n - 4*n^2)*a(n)
%F A052719 a(n) = n!*A000245(n-2). - _R. J. Mathar_, Oct 26 2013
%F A052719 From _G. C. Greubel_, May 28 2022: (Start)
%F A052719 G.f.: 6*x^3*Hypergeometric2F0([2, 3/2], [], 4*x).
%F A052719 E.g.f.: (1/4)*(1 + 2*x - 8*x^2 - (1 + 2*x)*sqrt(1-4*x)). (End)
%p A052719 spec := [S,{B=Union(Z,C),C=Prod(B,B),S=Prod(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052719 Table[If[n<2, 0, 3*(n-2)*(n-1)!*CatalanNumber[n-2]], {n,0,30}] (* _G. C. Greubel_, May 28 2022 *)
%o A052719 (SageMath) [0,0]+[3*(n-2)*factorial(n-1)*catalan_number(n-2) for n in (2..30)] # _G. C. Greubel_, May 28 2022
%Y A052719 Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052720, A052721, A052722, A052723.
%Y A052719 Cf. A000108, A000245.
%K A052719 easy,nonn
%O A052719 0,4
%A A052719 encyclopedia(AT)pommard.inria.fr, Jan 25 2000