A386789 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).

1, 0, 2, 0, 3, 6, 0, 4, 20, 20, 0, 5, 45, 105, 70, 0, 6, 84, 336, 504, 252, 0, 7, 140, 840, 2100, 2310, 924, 0, 8, 216, 1800, 6600, 11880, 10296, 3432, 0, 9, 315, 3465, 17325, 45045, 63063, 45045, 12870, 0, 10, 440, 6160, 40040, 140140, 280280, 320320, 194480, 48620

Offset: 0

Author

Peter Luschny, Aug 06 2025

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  [0] 1;
  [1] 0, 2;
  [2] 0, 3,   6;
  [3] 0, 4,  20,   20;
  [4] 0, 5,  45,  105,   70;
  [5] 0, 6,  84,  336,  504,   252;
  [6] 0, 7, 140,  840, 2100,  2310,   924;
  [7] 0, 8, 216, 1800, 6600, 11880, 10296, 3432;
.
Seen as an array A(n, k) = binomial(n + k - 1, n)*binomial(n + 2*k, k):
  [0] 1, 2,   6,   20,    70,    252,     924, ... [A000984]
  [1] 0, 3,  20,  105,   504,   2310,   10296, ... [A000917]
  [2] 0, 4,  45,  336,  2100,  11880,   63063, ...
  [3] 0, 5,  84,  840,  6600,  45045,  280280, ...
  [4] 0, 6, 140, 1800, 17325, 140140, 1009008, ...
  [5] 0, 7, 216, 3465, 40040, 378378, 3118752, ...
  [6] 0, 8, 315, 6160, 84084, 917280, 8576568, ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Crossrefs

Cf. A176479 (row sums), A000984 (main diagonal), A181983 (alternating row sums), A386876 (central terms).

Cf. A000984, A000917, A357367.

Programs

Maple

T := (n, k) -> binomial(n - 1, k - 1)*binomial(n + k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);

Mathematica

A386789[n_, k_] := Binomial[n - 1, k - 1]*Binomial[n + k, k];
Table[A386789[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Aug 06 2025 *)

Formula

A357367(n, k) = n!*T(n, k).