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A052721 Expansion of e.g.f. x*(1-2*x)*(1 - 2*x - sqrt(1-4*x))/2 - x^3.

%I A052721 #30 May 29 2022 03:18:03
%S A052721 0,0,0,0,0,120,2880,70560,1935360,59875200,2075673600,79913433600,
%T A052721 3387499315200,156883562035200,7884404656128000,427447366714368000,
%U A052721 24869664972472320000,1545805113445232640000,102232975285590589440000
%N A052721 Expansion of e.g.f. x*(1-2*x)*(1 - 2*x - sqrt(1-4*x))/2 - x^3.
%H A052721 G. C. Greubel, <a href="/A052721/b052721.txt">Table of n, a(n) for n = 0..350</a>
%H A052721 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=677">Encyclopedia of Combinatorial Structures 677</a>
%F A052721 D-finite with recurrence: a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=120, a(6)=2880, (n+2)*a(n+2) = (6*n^2 + 8*n - 8)*a(n+1) + (40 + 44*n = 4*n^2 - 8*n^3)*a(n).
%F A052721 a(n) = 2*Pi^(-1/2)*4^(n-3)*Gamma(n-5/2)*n*(n-4) for n>3.  - _Mark van Hoeij_, Oct 30 2011
%F A052721 a(n) = n!*A002057(n-5). - _R. J. Mathar_, Oct 18 2013
%F A052721 From _G. C. Greubel_, May 28 2022: (Start)
%F A052721 G.f.: 4!*x*(d/dx)( x^5 * Hypergeometric2F0([2, 5/2], [], 4*x) ).
%F A052721 E.g.f.: (x/2)*(1 - 4*x + 2*x^2 - (1-2*x)*sqrt(1-4*x)). (End)
%p A052721 spec := [S,{C=Union(B,Z),B=Prod(C,C),S=Prod(B,B,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052721 Table[If[n<5, 0, 2*n*(n-2)!*(n-4)*CatalanNumber[n-3]], {n,0,30}] (* _G. C. Greubel_, May 28 2022 *)
%o A052721 (SageMath)
%o A052721 def A052721(n):
%o A052721     if (n<5): return 0
%o A052721     else: return 2*n*factorial(n-2)*(n-4)*catalan_number(n-3)
%o A052721 [A052721(n) for n in (0..30)] # _G. C. Greubel_, May 28 2022
%Y A052721 Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052722, A052723.
%Y A052721 Cf. A000108, A002057.
%K A052721 easy,nonn
%O A052721 0,6
%A A052721 encyclopedia(AT)pommard.inria.fr, Jan 25 2000