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%I #22 Mar 17 2024 10:20:56
%S 1,3,22,225,2706,35861,507060,7510005,115175530,1815002145,
%T 29231242206,479251119815,7975209124260,134398986236625,
%U 2289535943534920,39370761619959165,682603570436824602,11921040322642855193
%N a(n) = (1/n)*Sum_{k=0..n} binomial(n, k)*binomial(n+k, k+1)*binomial(n+k, k) with a(0) = 1.
%H G. C. Greubel, <a href="/A073530/b073530.txt">Table of n, a(n) for n = 0..770</a>
%F a(n) = hypergeometric3F2([n+1, n+1, -n], [1, 2], -1).
%F Recurrence: 2*(n-1)*n*(n+1)*(59*n^2 - 235*n + 216)*a(n) = 3*(n-1)*(767*n^4 - 3822*n^3 + 6065*n^2 - 3602*n + 720)*a(n-1) + (n-2)*(n-1)*(295*n^3 - 1470*n^2 + 2051*n - 792)*a(n-2) + (n-3)^2*(n-2)*(59*n^2 - 117*n + 40)*a(n-3). - _Vaclav Kotesovec_, Mar 02 2014
%F a(n) ~ c * d^n/n^2, where d = 1/6*(39 + (61128 - 177*sqrt(177))^(1/3) + (3*(20376 + 59*sqrt(177)))^(1/3)) = 19.62866250831184052... is the root of the equation 2*d^3 - 39*d^2 - 5*d = 1 and c = sqrt(1/2 + sqrt(231/59) * cosh(arccosh(51 * sqrt(177/77)/77)/3)/2)/Pi = 0.38852216850573971010943128486103656013508... - _Vaclav Kotesovec_, Mar 02 2014, updated Mar 17 2024
%p p3 := x^3+5*x^2+39*x-2; p4 := x^4+4*x^3+30*x^2-20*x+1;
%p y := hypergeom([1/12, 5/12], [1], -1728*p3*x^4/p4^3)/p4^(1/4);
%p a1 := p3/(5*x^2+8*x); a2 := (13*x^3-197*x^2-60*x+16)/(5*x^2+8*x)^2;
%p ogf := a1*y - Int(a2*y,x) - 89/32;
%p series(ogf,x=0,20); # _Mark van Hoeij_, Apr 03 2013
%t Table[ HypergeometricPFQ[{n+1,n+1,-n}, {1,2}, -1], {n,0,20}] (* _Robert G. Wilson v_ *)
%o (Magma)
%o A073530:= func< n | n eq 0 select 1 else (1/n)*(&+[Binomial(n,j)* Binomial(n+j,j+1)*Binomial(n+j,j): j in [0..n]]) >;
%o [A073530(n): n in [0..30]]; // _G. C. Greubel_, Dec 27 2022
%o (SageMath)
%o def A073530(n):
%o if (n==0): return 1
%o else: return sum(binomial(n,j)*binomial(n+j,j+1)*binomial(n+j,j) for j in range(n+1))/n
%o [A073530(n) for n in range(31)] # _G. C. Greubel_, Dec 27 2022
%Y Cf. A074635, A074649.
%K nonn
%O 0,2
%A _Karol A. Penson_, Aug 29 2002
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