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A052722 Expansion of e.g.f. (1 - 2*x - sqrt(1-4*x))^2 * (1 - sqrt(1-4*x))/8.

%I A052722 #23 May 29 2022 03:17:21
%S A052722 0,0,0,0,0,120,3600,100800,3024000,99792000,3632428800,145297152000,
%T A052722 6351561216000,301699157760000,15487223431680000,854894733428736000,
%U A052722 50516506975334400000,3182539939446067200000,212985365178313728000000
%N A052722 Expansion of e.g.f. (1 - 2*x - sqrt(1-4*x))^2 * (1 - sqrt(1-4*x))/8.
%H A052722 G. C. Greubel, <a href="/A052722/b052722.txt">Table of n, a(n) for n = 0..350</a>
%H A052722 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=678">Encyclopedia of Combinatorial Structures 678</a>
%F A052722 D-finite with recurrence: a(0) = a(1) = a(2) = a(3) = a(4) = 0, a(5)=120, a(n+3) = (9+7*n)*a(n+2) + (14 - 19*n - 13*n^2)*a(n+1) - (20 + 22*n - 2*n^2 - 4*n^3)*a(n).
%F A052722 a(n) = n!*A000344(n-3). - _R. J. Mathar_, Oct 18 2013
%F A052722 From _G. C. Greubel_, May 28 2022: (Start)
%F A052722 G.f.: 5!*x^5*hypergeometric2F0([5/2, 3], [], 4*x).
%F A052722 E.g.f.: (1/2)*(1 - 5*x + 5*x^2 - (1 - 3*x + x^2)*sqrt(1-4*x)). (End)
%p A052722 spec := [S,{C=Union(B,Z),B=Prod(C,C),S=Prod(B,B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052722 With[{nn=20},CoefficientList[Series[((1-2x-Sqrt[1-4x])^2 (1-Sqrt[1-4x]))/8,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 30 2021 *)
%t A052722 Table[If[n<5, 0, 10*(n-2)!*Binomial[n-3,2]*CatalanNumber[n-3]], {n,0,30}] (* _G. C. Greubel_, May 28 2022 *)
%o A052722 (SageMath)
%o A052722 def A052722(n):
%o A052722     if (n<5): return 0
%o A052722     else: return 10*factorial(n-2)*binomial(n-3,2)*catalan_number(n-3)
%o A052722 [A052722(n) for n in (0..30)] # _G. C. Greubel_, May 28 2022
%Y A052722 Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052723.
%Y A052722 Cf. A000108, A000344.
%K A052722 easy,nonn
%O A052722 0,6
%A A052722 encyclopedia(AT)pommard.inria.fr, Jan 25 2000