|
%I #21 Oct 01 2018 03:09:54
%S 1,3,6,30,70,420,1050,6930,18018,126126,336336,2450448,6651216,
%T 49884120,137181330,1051723530,2921454250,22787343150,63804560820,
%U 504636071940,1422156202740,11377249621920,32235540595440,260363981732400
%N Number of permutations of 0..floor((n*3-2)/2) on odd squares of an n X 3 array such that each row and column of odd squares is increasing.
%C a(n) is number of symmetric standard Young tableaux of shape (n,n,n). - _Ran Pan_, May 21 2015
%H R. H. Hardin, <a href="/A215294/b215294.txt">Table of n, a(n) for n = 1..210</a>
%H Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/problems/p4.html">Problem 4</a>, Project P.
%F f3 = floor((n+1)/2), f4 = floor(n/2);
%F a(n) = A060854(1,f3)*A060854(2,f4)*binomial(1*f3+2*f4,1*f3).
%F a(n) = e(n) if n even otherwise o(n), where e(n) = 6*Gamma((3*n)/2))/((2 + n)*Gamma(1 + n/2)^2* Gamma(n/2)) and o(n) = ((1 + n)*Gamma(1/2 + (3*n)/2))/(2*Gamma((3 + n)/2)^3). - _Peter Luschny_, Sep 30 2018
%e Some solutions for n=5:
%e x 1 x x 0 x x 0 x x 4 x x 0 x x 1 x x 1 x
%e 0 x 5 2 x 4 2 x 5 0 x 2 1 x 2 0 x 5 0 x 3
%e x 3 x x 1 x x 1 x x 5 x x 3 x x 2 x x 2 x
%e 2 x 6 3 x 6 3 x 6 1 x 3 4 x 6 3 x 6 4 x 5
%e x 4 x x 5 x x 4 x x 6 x x 5 x x 4 x x 6 x
%p a := n -> `if`(irem(n, 2) = 0, ((1/2)*n+1)*factorial((3/2)*n)/ (factorial((1/2)*n+1)^2*factorial((1/2)*n)), factorial((3/2)*n+3/2)/ (factorial((1/2)*n+1/2)^3*((9/2)*n+3/2))): # _Peter Luschny_, Sep 30 2018
%Y Column 3 of A215297.
%Y Cf. A060693.
%K nonn
%O 1,2
%A _R. H. Hardin_, Aug 07 2012
| 