OFFSET
0,2
FORMULA
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)).
a(n) = (i/Pi)*Integral_{t=0..1} hypergeom([n, -n],[1], -8*t)/sqrt(t*(t-1)).
a(n) ~ 3*sqrt(2) * (1 + sqrt(2))^(4*n) / (8*Pi*n). - Vaclav Kotesovec, Jan 08 2024
MAPLE
a := n -> hypergeom([n, -n, 1/2], [1, 1], -8):
seq(simplify(a(n)), n = 0..17);
# Alternative:
A358388 := proc(n) local a;
a := proc(n) option remember; if n < 3 then return [1, 1, 9][n + 1] fi;
((n - 3)^2*(2*n - 3)*a(n - 3) - (35*(n - 4)*n + 131)*((2*n - 5)*a(n - 2)
+ (3 - 2*n)*a(n - 1))) / ((n - 1)^2*(2*n - 5)) end:
(a(n + 1) + a(n)) / 2 end: seq(A358388(n), n = 0..17);
MATHEMATICA
a[n_] := HypergeometricPFQ[{n, -n, 1/2}, {1, 1}, -8];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Nov 27 2023 *)
PROG
(Python)
def A358388gen() -> Generator:
c, b, a, n = 1, 1, 9, 2
while True:
yield (c + b) // 2
n += 1
f = 35 * (n - 4) * n + 131
aa = a * f * (2 * n - 3)
bb = b * f * (2 * n - 5)
cc = c * (n - 3) ** 2 * (2 * n - 3)
d = (aa - bb + cc) // ((n - 1) ** 2 * (2 * n - 5))
c, b, a = b, a, d
A358388 = A358388gen()
print([next(A358388) for n in range(18)])
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Nov 13 2022
STATUS
approved