cp's OEIS Frontend

A364361 Table read by rows. T(n, k) = Sum_{j=0..n-k} k*binomial(k, j)*binomial(n - j, k).

%I A364361 #8 Jul 30 2023 16:34:53
%S A364361 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 A364361 0,13,122,387,516,305,78,7,0,15,170,693,1284,1155,510,105,8,0,17,226,
%U A364361 1131,2724,3405,2262,791,136,9,0,19,290,1725,5156,8415,7734,4025,1160,171,10
%N A364361 Table read by rows. T(n, k) = Sum_{j=0..n-k} k*binomial(k, j)*binomial(n - j, k).
%F A364361 T(2*n, n) = n * LegendreP(n, 3).
%e A364361 The triangle begins:
%e A364361   [0] 0;
%e A364361   [1] 0,  1;
%e A364361   [2] 0,  3,   2;
%e A364361   [3] 0,  5,  10,   3;
%e A364361   [4] 0,  7,  26,   21,    4;
%e A364361   [5] 0,  9,  50,   75,   36,    5;
%e A364361   [6] 0, 11,  82,  189,  164,   55,    6;
%e A364361   [7] 0, 13, 122,  387,  516,  305,   78,   7;
%e A364361   [8] 0, 15, 170,  693, 1284, 1155,  510, 105,   8;
%e A364361   [9] 0, 17, 226, 1131, 2724, 3405, 2262, 791, 136, 9;
%e A364361 Seen as an array:
%e A364361   [0] 0,  1,   2,   3,     4,     5,      6,      7, ...  A001477
%e A364361   [1] 0,  3,  10,   21,   36,    55,     78,    105, ...  A014105
%e A364361   [2] 0,  5,  26,   75,  164,   305,    510,    791, ...  A048395
%e A364361   [3] 0,  7,  50,  189,  516,  1155,   2262,   4025, ...
%e A364361   [4] 0,  9,  82,  387, 1284,  3405,   7734,  15687, ...
%e A364361   [5] 0, 11, 122,  693, 2724,  8415,  21918,  50281, ...
%e A364361   [6] 0, 13, 170, 1131, 5156, 18265,  53934, 138775, ...
%e A364361   [7] 0, 15, 226, 1725, 8964, 35915, 118950, 340473, ...
%e A364361     A005408|A069894
%p A364361 T := (n, k) -> local j; add(k*binomial(k, j)*binomial(n-j, k), j = 0..n-k):
%p A364361 seq(seq(T(n, k), k = 0..n), n = 0..10);
%Y A364361 Cf. A364553 (row sums),  A364634 (main diagonal).
%Y A364361 Rows: A001477, A014105, A048395.
%Y A364361 Columns: A005408, A069894.
%K A364361 nonn,tabl
%O A364361 0,5
%A A364361 _Peter Luschny_, Jul 30 2023