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A362209 Irregular triangle read by rows: T(n, k) is the number of k X k matrices using all the integers from 1 to k^2 and having trace equal to n, with 1 <= k <= A003056(n).

%I A362209 #11 Apr 16 2023 20:37:44
%S A362209 1,0,0,4,0,4,0,8,0,4,4320,0,4,4320,0,0,8640,0,0,12960,0,0,17280,
%T A362209 11496038400,0,0,21600,11496038400,0,0,30240,22992076800,0,0,30240,
%U A362209 34488115200,0,0,34560,57480192000,0,0,34560,68976230400,291948240981196800000
%N A362209 Irregular triangle read by rows: T(n, k) is the number of k X k matrices using all the integers from 1 to k^2 and having trace equal to n, with 1 <= k <= A003056(n).
%F A362209 T(n, k) = A362187(k)*A362208(n, k).
%e A362209 Irregular triangle begins:
%e A362209     1;
%e A362209     0;
%e A362209     0, 4;
%e A362209     0, 4;
%e A362209     0, 8;
%e A362209     0, 4,  4320;
%e A362209     0, 4,  4320;
%e A362209     0, 0,  8640;
%e A362209     0, 0, 12960;
%e A362209     0, 0, 17280, 11496038400;
%e A362209     0, 0, 21600, 11496038400;
%e A362209     0, 0, 30240, 22992076800;
%e A362209     0, 0, 30240, 34488115200;
%e A362209     0, 0, 34560, 57480192000;
%e A362209     0, 0, 34560, 68976230400, 291948240981196800000;
%e A362209     ...
%e A362209 T(5,2) = 8 since we have:
%e A362209     [1, 2]  [1, 3]  [4, 2]  [4, 3]
%e A362209     [3, 4], [2, 4], [3, 1], [2, 1],
%e A362209 .
%e A362209     [2, 1]  [2, 4]  [3, 1]  [3, 4]
%e A362209     [4, 3], [1, 3], [4, 2], [1, 2].
%t A362209 A362208[n_,k_] := Length[Select[Join@@Permutations/@Select[IntegerPartitions[n, All, Range[k^2]], UnsameQ@@#&], Length[#]==k&]]; Table[(k^2-k)!A362208[n,k],{n,15},{k,Floor[(Sqrt[8n+1]-1)/2]}]//Flatten
%Y A362209 Cf. A000290, A003056 (row lengths), A345132, A362187, A362208.
%K A362209 nonn,tabf
%O A362209 1,4
%A A362209 _Stefano Spezia_, Apr 11 2023