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

1,3

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.

Links

Table of n, a(n) for n=1..55.

Formula

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

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

Example

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.

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

Prog

(PARI) T(n, k)={if(k<n, binomial(k+n-2, n-1), binomial(2*n-1, n))} \\ Andrew Howroyd, Nov 24 2024

Crossrefs

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

Sequence in context: A155734 A128162 A257253 * A067329 A170860 A170845

Adjacent sequences: A378373 A378375 A378376 * A378378 A378379 A378380

Keyword

nonn,tabl,new

Author

Zlatko Damijanic, Nov 24 2024

Status

approved