A259877 If n is even then a(n) = n!/( 2^(n/2)*(n/2)! ), otherwise a(n) = n!/( 3*2^((n-1)/2)*((n-3)/2)! ).

1, 1, 3, 10, 15, 105, 105, 1260, 945, 17325, 10395, 270270, 135135, 4729725, 2027025, 91891800, 34459425, 1964187225, 654729075, 45831035250, 13749310575, 1159525191825, 316234143225, 31623414322500, 7905853580625, 924984868933125, 213458046676875, 28887988983603750, 6190283353629375

Offset: 2

Author

N. J. A. Sloane, Jul 09 2015

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 2..501
• D. L. Andrews, Letter to N. J. A. Sloane, Apr 10 1978.
• D. L. Andrews and T. Thirunamachandran, On three-dimensional rotational averages, J. Chem. Phys., 67 (1977), 5026-5033. See N_n.
• D. L. Andrews and T. Thirunamachandran, On three-dimensional rotational averages, J. Chem. Phys., 67 (1977), 5026-5033. [Annotated scanned copy]

Crossrefs

A001147 alternating with A000457. Interlaced diagonal of A008299.

Programs

Maple

f:=proc(n) if n mod 2 = 0 then
n!/(2^(n/2)*(n/2)!) else
n!/( 3*2^((n-1)/2)*((n-3)/2)! ); fi; end;
[seq(f(n),n=2..30)];

Mathematica

Table[(n!/6)*2^(-n/2)*(((2^(1/2)*(1-(-1)^n))/(n/2-3/2)!)+3*(1+(-1)^n)/(n/2)!), {n, 2, 30}] (* Wesley Ivan Hurt, Jul 10 2015 *)

PARI

main(size)={v=vector(size);for(n=2, size+1, if(n%2==0, v[n-1]=n!/(2^(n/2)*(n/2)!), v[n-1]=n!/( 3*2^((n-1)/2)*((n-3)/2)!))); return(v);} /* Anders Hellström, Jul 10 2015 */

Python

from _future_ import division
A259877_list, a = [1], 1
for n in range(2,10**2):
    a = 6*a//(n-1) if n % 2 else a*n*(n+1)//6
    A259877_list.append(a) # Chai Wah Wu, Jul 15 2015

Formula

a(n) = (n!/6)*2^(-n/2)*(((2^(1/2)*(1-(-1)^n))/(n/2-3/2)!)+3*(1+(-1)^n)/(n/2)!). - Wesley Ivan Hurt, Jul 10 2015

a(n+1) = a(n)*n*(n+1)/6 if n is even, a(n+1) = 6*a(n)/(n-1) if n is odd. - Chai Wah Wu, Jul 15 2015

a(2*n) = A001147(n), a(2*n+1) = A000457(n-1). - Yuchun Ji, Nov 02 2020