A275329 a(n) = (2+[n/2])*n!/((1+[n/2])*[n/2]!^2).

2, 2, 3, 9, 8, 40, 25, 175, 84, 756, 294, 3234, 1056, 13728, 3861, 57915, 14300, 243100, 53482, 1016158, 201552, 4232592, 764218, 17577014, 2912168, 72804200, 11143500, 300874500, 42791040, 1240940160, 164812365, 5109183315, 636438060, 21002455980, 2463251010

Offset: 0

Author

Peter Luschny, Sep 10 2016

Keywords

• nonn

Crossrefs

Cf. A038665, A056040, A057977, A097070, A189911.

Programs

Maple

a := n -> (2+iquo(n,2))*n!/((1+iquo(n,2))*iquo(n, 2)!^2):
seq(a(n), n=0..34);

Sage

def A275329():
    x, n, k = 1, 1, 2
    while True:
        yield x * k
        if is_odd(n):
            x *= n
        else:
            k += 1
            x = (x<<2)//(n+2)
        n += 1
a = A275329(); print([next(a) for _ in range(37)])

Formula

a(n) = A056040(n)*(2+[n/2])/(1+[n/2]).

a(n) = A057977(n)*A008619(n+2).

a(2*n+1) = (n+2)*binomial(2*n+1, n+1) = A189911(2*n+1).

a(2*n-3) = n*binomial(2*n-3, n-1) = A097070(n) for n>=2.

a(2*n+2) = (n+3)*binomial(2*n+2, n+1)/(n+2) = A038665(n).

Sum_{n>=0} 1/a(n) = 16/3 - 40*Pi/(9*sqrt(3)) + 4*Pi^2/9. - Amiram Eldar, Aug 20 2022