A329965 a(n) = ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2.

1, 2, 6, 72, 240, 7200, 25200, 1411200, 5080320, 457228800, 1676505600, 221298739200, 821966745600, 149597947699200, 560992303872000, 134638152929280000, 508633022177280000, 155641704786247680000, 591438478187741184000, 224746621711341649920000

Offset: 0

Author

Peter Luschny, Dec 04 2019

Keywords

• nonn

Crossrefs

Cf. A212303, A057977, A052510, A123072, A093005, A262033, A329964.

Programs

Maple

A329965 := n -> ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2:
seq(A329965(n), n=0..19);

Mathematica

ser := Series[(1 - Sqrt[1 - 4 x^2] - 4 x^2 (1 - x - Sqrt[1 - 4 x^2]))/(2 x^2 (1 - 4 x^2)^(3/2)), {x, 0, 22}]; Table[n! Coefficient[ser, x, n], {n, 0, 20}]
Table[(1+n)Floor[1+n/2](n!/Floor[1+n/2]!)^2,{n,0,30}] (* Harvey P. Dale, Oct 01 2023 *)

Python

from fractions import Fraction
def A329965():
    x, n = 1, Fraction(1)
    while True:
        yield int(x)
        m = n if n % 2 else 4/(n+2)
        n += 1
        x *= m * n
a = A329965(); [next(a) for i in range(36)]

Formula

a(n) = n!*A212303(n+1).

a(n) = (n+1)!*A057977(n).

a(n) = A093005(n+1)*A262033(n)^2.

a(n) = A093005(n+1)*A329964(n).

a(2*n) = A052510(n) (n >= 0).

a(2*n+1) = A123072(n+1) (n >= 0).

a(n) = n! [x^n] (1 - sqrt(1 - 4*x^2) - 4*x^2*(1 - x - sqrt(1 - 4*x^2)))/(2*x^2*(1 - 4*x^2)^(3/2)).