OFFSET
1,2
COMMENTS
Always an odd integer.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..345
FORMULA
a(n) = 2^(n-1)*((2*n)!/n!)*J(n) where J(n) = Integral_{0..Pi/3} sin(t)^(2*n-1) dt is given by the order 2 recursion : J(1) = 1/2, J(2) = 5/24, J(n) = 1/(8*n-4)*((14*n - 17)*J(n-1) - 6*(n-2)*J(n-1)).
From G. C. Greubel, Aug 04 2021: (Start)
a(n) = (1/4) * (3/2)^n * (n-1)! * binomial(2*n, n) * Hypergeometric2F1([1/2, n], [n+1], 3/4).
a(n) = (1/4) * (3/2)^n * n! * binomial(2*n, n) * Sum_{k>=0} binomial(2*k, k)*(3/16)^k/(n+k). (End)
MATHEMATICA
a[n_]:= Round[(3/2)^n*((n-1)!/4)*Binomial[2*n, n]*Hypergeometric2F1[1/2, n, n+1, 3/4]]; Table[a[n], {n, 40}] (* G. C. Greubel, Aug 04 2021 *)
PROG
(Sage)
@CachedFunction
def f(n): return (2*n+(-1)^n)/factorial(2*n) if (n<3) else 1/(4*(2*n-1))*((14*n - 17)*f(n-1) - 6*(n-2)*f(n-2))
def a(n): return 2^(n-1)*factorial(n)*binomial(2*n, n)*f(n)
[a(n) for n in (1..40)] # G. C. Greubel, Aug 04 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Sep 30 2006
STATUS
approved