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Expansion of e.g.f.: (1 - 4*x + 3*x^2)*(1 - 2*x - sqrt(1-4*x))/2 - x^2 + 2*x^3.
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%I #21 May 29 2022 03:18:54

%S 0,0,0,0,0,0,720,30240,1088640,39916800,1556755200,65383718400,

%T 2964061900800,144815595724800,7602818775552000,427447366714368000,

%U 25646842002862080000,1636734826000834560000,110752389892723138560000

%N Expansion of e.g.f.: (1 - 4*x + 3*x^2)*(1 - 2*x - sqrt(1-4*x))/2 - x^2 + 2*x^3.

%H G. C. Greubel, <a href="/A052720/b052720.txt">Table of n, a(n) for n = 0..350</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=676">Encyclopedia of Combinatorial Structures 676</a>

%F D-finite with recurrence: a(1)=0; a(2)=0; a(4)=0; a(3)=0; a(5)=0; a(6)=720; a(n+3) = (10+8*n)*a(n+2) + (22-27*n-19*n^2)*a(n+1) - (60-66*n+6*n^2+12*n^3)*a(n).

%F a(n) = n!*A003517(n-4). - _R. J. Mathar_, Oct 18 2013

%F From _G. C. Greubel_, May 28 2022: (Start)

%F G.f.: 6!*x^6*Hypergeometric2F0([3, 7/2], [], 4*x).

%F E.g.f.: (1/2)*(1 - 6*x + 9*x^2 - 2*x^3 - (1 - 4*x + 3*x^2)*sqrt(1-4*x)). (End)

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

%t Table[If[n<6, 0, 6*(n-2)!*Binomial[n-4, 2]*CatalanNumber[n-3]], {n,0,30}] (* _G. C. Greubel_, May 28 2022 *)

%o (SageMath)

%o def A052720(n):

%o if (n<6): return 0

%o else: return 6*factorial(n-2)*binomial(n-4,2)*catalan_number(n-3)

%o [A052720(n) for n in (0..30)] # _G. C. Greubel_, May 28 2022

%Y Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052721, A052722, A052723.

%Y Cf. A000108, A003517.

%K easy,nonn

%O 0,7

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