A237884 a(n) = (n!*m)/(m!*(m+1)!) where m = floor(n/2).

0, 0, 1, 3, 4, 20, 15, 105, 56, 504, 210, 2310, 792, 10296, 3003, 45045, 11440, 194480, 43758, 831402, 167960, 3527160, 646646, 14872858, 2496144, 62403600, 9657700, 260757900, 37442160, 1085822640, 145422675, 4508102925, 565722720, 18668849760, 2203961430

Offset: 0

Author

Peter Luschny, Feb 14 2014

Keywords

• nonn

Programs

Maple

A237884 := proc(n) m := iquo(n,2); (n!*m)/(m!*(m+1)!) end;
seq(A237884(n), n = 0..34);

Mathematica

CoefficientList[Series[-((-1 + Sqrt[1 - 4 x^2] -x (-1 + Sqrt[1 - 4 x^2] +
2 x (-3 + 2 Sqrt[1 - 4 x^2] +x (3 + 4 x - 2 Sqrt[1 - 4 x^2]))))/
(2 x^2 (1 - 4 x^2)^(3/2))), {x, 0, 30}], x] (* Benedict W. J. Irwin, Aug 15 2016 *)
Table[(n! #)/(#! (# + 1)!) &@ Floor[n/2], {n, 0, 34}] (* Michael De Vlieger, Aug 15 2016 *)

Sage

def A237884():
    r, s, n = 1, 0, 0
    while True:
        yield s
        n += 1
        r *= 4/n if is_even(n) else n
        s = r * (n//2)/(n//2+1)
a = A237884(); [next(a) for i in range(35)]

Formula

a(2*n) = A001791(n).

a(2*n+1) = A000917(n-1).

a(n) = n^(n mod 2)*binomial(2*floor(n/2), floor(n/2)-1).

a(n) = A162246(n, n+2) = n!/((n-ceiling((n+2)/2))!*floor((n+2)/2)!) if n > 1, otherwise 0.

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

a(n) + A056040(n) = A057977(n).

G.f.: -((p - 1 - x*(p - 1 + 2*x*(2*p - 3 + x*(3 + 4*x - 2*p))))/(2*x^2*p^3)), where p=sqrt(1-4*x^2). - Benedict W. J. Irwin, Aug 15 2016