A123130 a(n) = 2^(n-1)*((2*n)!/n!) * Integral_{t=0..Pi/3} sin(t)^(2*n-1) dt.

1, 5, 53, 867, 19239, 539925, 18338445, 731412675, 33511100175, 1734534350325, 100101650876325, 6373296156687075, 443776641732321975, 33548286541938693525, 2736444872641087532925, 239549584572054489607875

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

Sequence in context: A118583 A090360 A367156 * A094089 A357343 A231866

Adjacent sequences: A123127 A123128 A123129 * A123131 A123132 A123133

Keyword

nonn

Author

Benoit Cloitre, Sep 30 2006

Status

approved