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