A242986 a(n) = 6*(n+1)!/((3+floor(n/2))*(floor(n/2)!)^2).

2, 4, 9, 36, 36, 216, 140, 1120, 540, 5400, 2079, 24948, 8008, 112112, 30888, 494208, 119340, 2148120, 461890, 9237800, 1790712, 39395664, 6953544, 166885056, 27041560, 703080560, 105306075, 2948570100, 410605200, 12318156000, 1602881040, 51292193280, 6263890380

Offset: 0

Author

Peter Luschny, Aug 25 2014

Keywords

• nonn

Crossrefs

Cf. A007946, A056040.

Programs

Magma

[6*Factorial(n+1)/((3+Floor(n/2))*(Factorial(Floor(n/2)))^2) : n in [0..30]]; // Wesley Ivan Hurt, Aug 26 2014

Maple

A056040 := n -> n!/iquo(n,2)!^2;
A242986 := n -> (6*(n+1)/(3+iquo(n,2)))*A056040(n);
seq(A242986(n), n=0..32);

Mathematica

Table[6(n + 1)!/((3 + Floor[n/2])*(Floor[n/2]!)^2), {n, 0, 30}] (* Wesley Ivan Hurt, Aug 26 2014 *)

Sage

@CachedFunction
def A242986(n):
    if n == 0: return 2
    h = (n+1)*A242986(n-1)
    if 2.divides(n):
        h *= (4*(n+4))/(n^2*(n+6))
    return h
[A242986(n) for n in range(33)]

Formula

a(n) = (6*(n+1)/(3+floor(n/2)))*A056040(n).

a(2*n) = A007946(n).

Recurrence: a(n) = a(n-1)*(n+1)*(4*(n+4))/(n^2*(n+6)) if n mod 2 = 0 else a(n-1)*(n+1) for n>0, a(0) = 2.

Asymptotic: a(x) ~ exp(x*log(2) - log(Pi)/2 - cos(Pi*x)*(log(x/2) + 1/(2*x))/2 + log(6*(x+1)) - log(3+floor(x/2))) for x>=1.

G.f.: (4*x-1)/(2*x^6) + (-16*x^7+16*x^6-48*x^5+12*x^4+48*x^3-12*x^2-8*x+2)/(4*(1-4*x^2)^(3/2)*x^6). - Robert Israel, Aug 25 2014

Sum_{n>=0} 1/a(n) = Pi^2/54 + 19*Pi/(54*sqrt(3)) + 1/9. - Amiram Eldar, Feb 17 2023