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

0,5

Links

Table of n, a(n) for n=0..65.

Formula

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

Example

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

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);

Crossrefs

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

Rows: A001477, A014105, A048395.

Columns: A005408, A069894.

Sequence in context: A326296 A246773 A359843 * A338022 A253176 A079408

Adjacent sequences: A364358 A364359 A364360 * A364362 A364363 A364364

Keyword

nonn,tabl

Author

Peter Luschny, Jul 30 2023

Status

approved