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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).
3
1, 0, 0, 4, 0, 4, 0, 8, 0, 4, 4320, 0, 4, 4320, 0, 0, 8640, 0, 0, 12960, 0, 0, 17280, 11496038400, 0, 0, 21600, 11496038400, 0, 0, 30240, 22992076800, 0, 0, 30240, 34488115200, 0, 0, 34560, 57480192000, 0, 0, 34560, 68976230400, 291948240981196800000
OFFSET
1,4
FORMULA
T(n, k) = A362187(k)*A362208(n, k).
EXAMPLE
Irregular triangle begins:
1;
0;
0, 4;
0, 4;
0, 8;
0, 4, 4320;
0, 4, 4320;
0, 0, 8640;
0, 0, 12960;
0, 0, 17280, 11496038400;
0, 0, 21600, 11496038400;
0, 0, 30240, 22992076800;
0, 0, 30240, 34488115200;
0, 0, 34560, 57480192000;
0, 0, 34560, 68976230400, 291948240981196800000;
...
T(5,2) = 8 since we have:
[1, 2] [1, 3] [4, 2] [4, 3]
[3, 4], [2, 4], [3, 1], [2, 1],
.
[2, 1] [2, 4] [3, 1] [3, 4]
[4, 3], [1, 3], [4, 2], [1, 2].
MATHEMATICA
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
CROSSREFS
Cf. A000290, A003056 (row lengths), A345132, A362187, A362208.
Sequence in context: A098002 A241658 A256719 * A343722 A035622 A112919
KEYWORD
nonn,tabf
AUTHOR
Stefano Spezia, Apr 11 2023
STATUS
approved