A299506 a(n) = hypergeom([-n, n - 1/2], [1], -4).

1, 3, 43, 661, 10515, 171097, 2828101, 47284251, 797456947, 13540982665, 231188344401, 3964874384863, 68252711769373, 1178662654873191, 20409993947488075, 354260920943874245, 6161735337225790035, 107368528677807960185, 1873946997372948997345, 32754419073618998202975

Offset: 0

Author

Peter Luschny, Mar 16 2018

Keywords

• nonn

Crossrefs

Cf. A299845, A243946, A084769, A243947.

Cf. A300945, A300946.

Programs

Mathematica

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

Formula

From Vaclav Kotesovec, Jul 05 2018: (Start)

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).

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)

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