A378377 Triangle read by rows: T(n,k) is the number of non-descending sequences with length k such that the maximum of the length and the last number is n.

1, 1, 3, 1, 3, 10, 1, 4, 10, 35, 1, 5, 15, 35, 126, 1, 6, 21, 56, 126, 462, 1, 7, 28, 84, 210, 462, 1716, 1, 8, 36, 120, 330, 792, 1716, 6435, 1, 9, 45, 165, 495, 1287, 3003, 6435, 24310, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 92378

Offset: 1

Author

Zlatko Damijanic, Nov 24 2024

Keywords

• nonn
• tabl

Comments

Also the T(n,k) is the number of integer partitions (of any positive integer) with length k such that the maximum of the length and the largest part is n.

When k < n, then the last number is n.

Examples

			Triangle begins:
  1
  1 3
  1 3 10
  1 4 10 35
  1 5 15 35 126
  1 6 21 56 126 462
  1 7 28 84 210 462 1716
  ...
For T(3,1) solution is |{(3)}| = 1.
For T(3,2) solution is |{(1,3), (2,3), (3,3)}| = 3.
For T(3,3) solution is |{(1,1,1), (1,1,2), (1,1,3), (1,2,2), (1,2,3), (1,3,3), (2,2,2), (2,2,3), (2,3,3), (3,3,3)}| = 10.
		

Crossrefs

Cf. A051924 (row sums), A001700 (right diagonal).

Programs

Mathematica

T[n_, k_] := Which[
  k == 1, 1,
  k == n, Binomial[2n-1, n],
  k == n-1, T[n-1, n-1],
  1 < k < n-1, T[n-1, k] + T[n, k-1]
];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten

PARI

T(n,k)={if(kAndrew Howroyd, Nov 24 2024

Formula

T(n,n) = binomial(2*n-1,n).

T(n,k) = binomial(k+n-2, n-1) for k < n.