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Number of 3*n X 3 0..2 arrays with row sums 2 and column sums 2*n.
1

%I #23 May 17 2023 13:01:40

%S 21,2385,352128,57775905,10060071021,1820016119376,338183208699840,

%T 64089909936535329,12331175198408791725,2401214665364782652385,

%U 472159936393091112404160,93594776429965445731933200

%N Number of 3*n X 3 0..2 arrays with row sums 2 and column sums 2*n.

%C Apparently the sequence gives the even-order terms in the diagonal of the rational function R(x,y,z) = 1/(1 - (x^2 + y^2 + z^2 - x*y - y*z + x*z)), i.e., a(n) = [(x*y*z)^(2*n)] R(x,y,z), n >= 1. - _Gheorghe Coserea_, Aug 09 2018

%C This is because those even-order terms would be the same if y were replaced by -y, giving S(x,y,z) = 1/(1 - (x^2 + y^2 + z^2 + x*y + y*z + x*z)) = Sum_{i>=0} (x^2 + y^2 + z^2 + x*y + y*z + x*z)^i, and the terms for (x*y*z)^(2*k) come from i=3*k in this sum. - _Robert Israel_, Jan 15 2023

%H R. H. Hardin, <a href="/A172675/b172675.txt">Table of n, a(n) for n = 1..33</a>

%p f:= proc(n) coeftayl((x^2+y^2+z^2+x*y+y*z+x*z)^(3*n),[x,y,z]=[0,0,0],[2*n,2*n,2*n]) end proc:

%p map(f, [$1..30]); # _Robert Israel_, Jan 15 2023

%t a[n_] := SeriesCoefficient[(x^2 + y^2 + z^2 + x*y + y*z + x*z)^(3n), {x, 0, 2n}, {y, 0, 2n}, {z, 0, 2n}];

%t Table[a[n], {n, 1, 12}] (* _Jean-François Alcover_, May 17 2023, after _Robert Israel_ *)

%K nonn

%O 1,1

%A _R. H. Hardin_, Feb 06 2010