A364361 Table read by rows. T(n, k) = Sum_{j=0..n-k} k*binomial(k, j)*binomial(n - j, k).
0, 0, 1, 0, 3, 2, 0, 5, 10, 3, 0, 7, 26, 21, 4, 0, 9, 50, 75, 36, 5, 0, 11, 82, 189, 164, 55, 6, 0, 13, 122, 387, 516, 305, 78, 7, 0, 15, 170, 693, 1284, 1155, 510, 105, 8, 0, 17, 226, 1131, 2724, 3405, 2262, 791, 136, 9, 0, 19, 290, 1725, 5156, 8415, 7734, 4025, 1160, 171, 10
Offset: 0
Examples
The triangle begins:
[0] 0;
[1] 0, 1;
[2] 0, 3, 2;
[3] 0, 5, 10, 3;
[4] 0, 7, 26, 21, 4;
[5] 0, 9, 50, 75, 36, 5;
[6] 0, 11, 82, 189, 164, 55, 6;
[7] 0, 13, 122, 387, 516, 305, 78, 7;
[8] 0, 15, 170, 693, 1284, 1155, 510, 105, 8;
[9] 0, 17, 226, 1131, 2724, 3405, 2262, 791, 136, 9;
Seen as an array:
[0] 0, 1, 2, 3, 4, 5, 6, 7, ... A001477
[1] 0, 3, 10, 21, 36, 55, 78, 105, ... A014105
[2] 0, 5, 26, 75, 164, 305, 510, 791, ... A048395
[3] 0, 7, 50, 189, 516, 1155, 2262, 4025, ...
[4] 0, 9, 82, 387, 1284, 3405, 7734, 15687, ...
[5] 0, 11, 122, 693, 2724, 8415, 21918, 50281, ...
[6] 0, 13, 170, 1131, 5156, 18265, 53934, 138775, ...
[7] 0, 15, 226, 1725, 8964, 35915, 118950, 340473, ...
A005408|A069894
Crossrefs
Programs
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Maple
T := (n, k) -> local j; add(k*binomial(k, j)*binomial(n-j, k), j = 0..n-k): seq(seq(T(n, k), k = 0..n), n = 0..10);
Formula
T(2*n, n) = n * LegendreP(n, 3).