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A358388 a(n) = hypergeom([n, -n, 1/2], [1, 1], -8).

%I A358388 #22 Jan 08 2024 05:19:07
%S A358388 1,5,89,2069,53505,1467765,41817305,1223277221,36488826881,
%T A358388 1104851215205,33853917808089,1047387818876085,32664869254856961,
%U A358388 1025606670801743061,32387641973278794585,1027864812983545977669,32762392278424747311105,1048268251830512324353221
%N A358388 a(n) = hypergeom([n, -n, 1/2], [1, 1], -8).
%F A358388 a(n) = Sum_{k=0..n} 2^(k - 1) * binomial(2*k, k)^2 * (binomial(n + k, 2*k) + binomial(n + k - 1, 2*k)).
%F A358388 a(n) = (i/Pi)*Integral_{t=0..1} hypergeom([n, -n],[1], -8*t)/sqrt(t*(t-1)).
%F A358388 a(n) ~ 3*sqrt(2) * (1 + sqrt(2))^(4*n) / (8*Pi*n). - _Vaclav Kotesovec_, Jan 08 2024
%p A358388 a := n -> hypergeom([n, -n, 1/2], [1, 1], -8):
%p A358388 seq(simplify(a(n)), n = 0..17);
%p A358388 # Alternative:
%p A358388 A358388 := proc(n) local a;
%p A358388 a := proc(n) option remember; if n < 3 then return [1, 1, 9][n + 1] fi;
%p A358388 ((n - 3)^2*(2*n - 3)*a(n - 3) - (35*(n - 4)*n + 131)*((2*n - 5)*a(n - 2)
%p A358388 + (3 - 2*n)*a(n - 1))) / ((n - 1)^2*(2*n - 5)) end:
%p A358388 (a(n + 1) + a(n)) / 2 end: seq(A358388(n), n = 0..17);
%t A358388 a[n_] := HypergeometricPFQ[{n, -n, 1/2}, {1, 1}, -8];
%t A358388 Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Nov 27 2023 *)
%o A358388 (Python)
%o A358388 def A358388gen() -> Generator:
%o A358388     c, b, a, n = 1, 1, 9, 2
%o A358388     while True:
%o A358388         yield (c + b) // 2
%o A358388         n += 1
%o A358388         f = 35 * (n - 4) * n + 131
%o A358388         aa = a * f * (2 * n - 3)
%o A358388         bb = b * f * (2 * n - 5)
%o A358388         cc = c * (n - 3) ** 2 * (2 * n - 3)
%o A358388         d = (aa - bb + cc) // ((n - 1) ** 2 * (2 * n - 5))
%o A358388         c, b, a = b, a, d
%o A358388 A358388 = A358388gen()
%o A358388 print([next(A358388) for n in range(18)])
%Y A358388 Cf. A001850, A243949, A358387.
%K A358388 nonn
%O A358388 0,2
%A A358388 _Peter Luschny_, Nov 13 2022