A364901 The n-volume of the unit regular n-simplex is sqrt(A364900(n))/a(n), with A364900(n) being squarefree.

1, 1, 4, 12, 96, 480, 5760, 20160, 215040, 5806080, 116121600, 1277337600, 30656102400, 398529331200, 11158821273600, 83691159552000, 5356234211328000, 30351993864192000, 3278015337332736000, 62282291409321984000, 2491291656372879360000, 52317124783830466560000

Offset: 0

Author

Jianing Song, Aug 12 2023

Keywords

• nonn
• easy

Examples

			  n |  the n-volume of the
    | unit regular n-simplex
  2 |  sqrt(3)/4 = A120011
  3 |  sqrt(2)/12 = A020829
  4 |  sqrt(5)/96 = A364895
  5 |  sqrt(3)/480
  6 |  sqrt(7)/5760
  7 |        1/20160
  8 |        1/215040
  9 |  sqrt(5)/5806080
		

Links

• Jianing Song, Table of n, a(n) for n = 0..425
• Wikipedia, Simplex

Crossrefs

Cf. A000188, A364900, A120011, A020829, A364895.

Programs

PARI

A000188(n) = sqrtint(n/core(n));
a(n) = n! * if(n%2, 2^((n-1)/2)/A000188((n+1)/2), 2^(n/2)/A000188(n+1))

Formula

The n-volume of the unit regular n-simplex is sqrt(n+1)/(n!*2^(n/2)), so a(n) = n! * 2^(n/2) / A000188(n+1) for even n and n! * 2^((n-1)/2) / A000188((n+1)/2) for odd n. It's easy to see that a(n) is an integer.