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

%I A364901 #17 Aug 20 2023 10:51:41
%S A364901 1,1,4,12,96,480,5760,20160,215040,5806080,116121600,1277337600,
%T A364901 30656102400,398529331200,11158821273600,83691159552000,
%U A364901 5356234211328000,30351993864192000,3278015337332736000,62282291409321984000,2491291656372879360000,52317124783830466560000
%N A364901 The n-volume of the unit regular n-simplex is sqrt(A364900(n))/a(n), with A364900(n) being squarefree.
%H A364901 Jianing Song, <a href="/A364901/b364901.txt">Table of n, a(n) for n = 0..425</a>
%H A364901 Wikipedia, <a href="https://en.wikipedia.org/wiki/Simplex">Simplex</a>
%F A364901 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.
%e A364901   n |  the n-volume of the
%e A364901     | unit regular n-simplex
%e A364901   2 |  sqrt(3)/4 = A120011
%e A364901   3 |  sqrt(2)/12 = A020829
%e A364901   4 |  sqrt(5)/96 = A364895
%e A364901   5 |  sqrt(3)/480
%e A364901   6 |  sqrt(7)/5760
%e A364901   7 |        1/20160
%e A364901   8 |        1/215040
%e A364901   9 |  sqrt(5)/5806080
%o A364901 (PARI) A000188(n) = sqrtint(n/core(n));
%o A364901 a(n) = n! * if(n%2, 2^((n-1)/2)/A000188((n+1)/2), 2^(n/2)/A000188(n+1))
%Y A364901 Cf. A000188, A364900, A120011, A020829, A364895.
%K A364901 nonn,easy
%O A364901 0,3
%A A364901 _Jianing Song_, Aug 12 2023