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a(n) = hypergeom([-n, n - 1/2], [1], -4).
2

%I #12 May 22 2024 09:18:37

%S 1,3,43,661,10515,171097,2828101,47284251,797456947,13540982665,

%T 231188344401,3964874384863,68252711769373,1178662654873191,

%U 20409993947488075,354260920943874245,6161735337225790035,107368528677807960185,1873946997372948997345,32754419073618998202975

%N a(n) = hypergeom([-n, n - 1/2], [1], -4).

%F From _Vaclav Kotesovec_, Jul 05 2018: (Start)

%F Recurrence: n*(2*n - 3)*(4*n - 7)*a(n) = 9*(4*n - 5)*(4*n^2 - 10*n + 5)*a(n-1) - (n-1)*(2*n - 5)*(4*n - 3)*a(n-2).

%F a(n) ~ 2^(-3/2) * sqrt(5) * phi^(6*n - 3/2) / sqrt(Pi*n), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. (End)

%F a(n) = 4^n*Sum_{k=0..n} (5/4)^k*(Gamma(n + 1)*Gamma(n - 1/2))/(Gamma(k + 1)*Gamma(n - k + 1)^2*Gamma(k - 1/2)). - _Detlef Meya_, May 22 2024

%t a[n_] := Hypergeometric2F1[-n, n - 1/2, 1, -4]; Table[a[n], {n, 0, 19}]

%t a[n_] := 4^n*Sum[(5/4)^k*(Gamma[n + 1]*Gamma[n - 1/2])/(Gamma[k + 1]*Gamma[n - k + 1]^2*Gamma[k - 1/2]),{k,0,n}]; Flatten[Table[a[n],{n,0,19}]] (* _Detlef Meya_, May 22 2024 *)

%Y Cf. A299845, A243946, A084769, A243947.

%Y Cf. A300945, A300946.

%K nonn

%O 0,2

%A _Peter Luschny_, Mar 16 2018