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Table read by rows. T(n, k) = Sum_{j=0..n-k} k*binomial(k, j)*binomial(n - j, k).
2

%I #8 Jul 30 2023 16:34:53

%S 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,

%T 0,13,122,387,516,305,78,7,0,15,170,693,1284,1155,510,105,8,0,17,226,

%U 1131,2724,3405,2262,791,136,9,0,19,290,1725,5156,8415,7734,4025,1160,171,10

%N Table read by rows. T(n, k) = Sum_{j=0..n-k} k*binomial(k, j)*binomial(n - j, k).

%F T(2*n, n) = n * LegendreP(n, 3).

%e The triangle begins:

%e [0] 0;

%e [1] 0, 1;

%e [2] 0, 3, 2;

%e [3] 0, 5, 10, 3;

%e [4] 0, 7, 26, 21, 4;

%e [5] 0, 9, 50, 75, 36, 5;

%e [6] 0, 11, 82, 189, 164, 55, 6;

%e [7] 0, 13, 122, 387, 516, 305, 78, 7;

%e [8] 0, 15, 170, 693, 1284, 1155, 510, 105, 8;

%e [9] 0, 17, 226, 1131, 2724, 3405, 2262, 791, 136, 9;

%e Seen as an array:

%e [0] 0, 1, 2, 3, 4, 5, 6, 7, ... A001477

%e [1] 0, 3, 10, 21, 36, 55, 78, 105, ... A014105

%e [2] 0, 5, 26, 75, 164, 305, 510, 791, ... A048395

%e [3] 0, 7, 50, 189, 516, 1155, 2262, 4025, ...

%e [4] 0, 9, 82, 387, 1284, 3405, 7734, 15687, ...

%e [5] 0, 11, 122, 693, 2724, 8415, 21918, 50281, ...

%e [6] 0, 13, 170, 1131, 5156, 18265, 53934, 138775, ...

%e [7] 0, 15, 226, 1725, 8964, 35915, 118950, 340473, ...

%e A005408|A069894

%p T := (n, k) -> local j; add(k*binomial(k, j)*binomial(n-j, k), j = 0..n-k):

%p seq(seq(T(n, k), k = 0..n), n = 0..10);

%Y Cf. A364553 (row sums), A364634 (main diagonal).

%Y Rows: A001477, A014105, A048395.

%Y Columns: A005408, A069894.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Jul 30 2023