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For n > 0, 0 <= k <= n^2, T(n,k) is the number of rotationally and reflectively distinct n X n arrays that contain the numbers 1 through k once each and n^2-k zeros.
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%I #15 Aug 13 2015 19:13:45

%S 1,1,1,1,2,3,3,1,3,12,66,378,1890,7560,22680,45360,45360,1,3,33,426,

%T 5466,65520,720720,7207200,64864800,518918400,3632428800,21794572800,

%U 108972864000,435891456000,1307674368000,2615348736000,2615348736000,1

%N For n > 0, 0 <= k <= n^2, T(n,k) is the number of rotationally and reflectively distinct n X n arrays that contain the numbers 1 through k once each and n^2-k zeros.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Array.html">Array</a>

%F T(n, k) = 1/8*([n^2]_k+2*[n]_k+5*[0]_k) if n is even and 1/8*([n^2]_k+4*[n]_k+3*[1]_k) if n is odd, where [m]_k=m*(m-1)*...*(m-k+1), k>0, [m]_0=1, is falling factorial. - _Vladeta Jovovic_, Aug 10 2003

%e 1,1; 1,1,2,3,3; 1,3,12,66,378,1890,7560,22680,45360,45360; ...

%e There is a single distinct 3 X 3 matrix containing all zeros, so a(3,1)=1.

%e There are 3 distinct 3 X 3 matrices containing a 1 and otherwise 0's, so a(3,2)=3.

%e There are 12 distinct 3 X 3 matrices containing a single 1, a single 2 and otherwise 0's, so a(3,3)=12.

%e There is a single distinct 3 X 3 matrix containing all zeros, so a(3, 0) = 1.

%e There are 3 distinct 3 X 3 matrices containing 8 0's and a 1, so a(3, 1) = 3.

%e There are 12 distinct 3 X 3 matrices containing a single 1, a single 2 and otherwise 0's, so a(3, 2) = 12.

%t (For 3 X 3 case) CanonicalizeArray[ x_ ] := Module[ {r, t}, Sort[ {x, Reverse[ x ], r=Reverse/@x, Reverse[ r ], t=Transpose[ x ], Reverse[ t ], r=Reverse/@t, Reverse[ r ]} ][ [ 1 ] ] ] ADD[ n_, d_ ] := Union[ CanonicalizeArray /@ (Partition[ #, n ] & /@ Permutations[ Join[ Range[ d ], Table[ 0, {n^2 - d} ] ] ]) ]

%Y Cf. A086829, A085698.

%K nonn,tabf

%O 1,5

%A _Zak Seidov_ and _Eric W. Weisstein_, Aug 08 2003

%E More terms from _David Wasserman_, Apr 12 2005