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

0,3

Links

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

Wikipedia, Simplex.

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.

Example

  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

Mathematica

f[p_, e_] := p^Floor[e/2]; a[0] = a[1] = 1; a[n_] := n! * If[EvenQ[n], 2^(n/2), 2^((n-1)/2)] / Times @@ f @@@ FactorInteger[If[EvenQ[n], n+1, (n+1)/2]]; Array[a, 22, 0] (* Amiram Eldar, Mar 18 2026 *)

Prog

(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))

Crossrefs

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

Sequence in context: A009651 A074930 A287596 * A268363 A377531 A038053

Adjacent sequences: A364898 A364899 A364900 * A364902 A364903 A364904

Keyword

nonn,easy

Author

Jianing Song, Aug 12 2023

Status

approved