A378377 - OEIS

%I #19 Dec 06 2024 11:18:29

%S 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,

%T 462,1716,1,8,36,120,330,792,1716,6435,1,9,45,165,495,1287,3003,6435,

%U 24310,1,10,55,220,715,2002,5005,11440,24310,92378

%N 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.

%C 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.

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

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

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

%e Triangle begins:

%e   1

%e   1 3

%e   1 3 10

%e   1 4 10 35

%e   1 5 15 35 126

%e   1 6 21 56 126 462

%e   1 7 28 84 210 462 1716

%e   ...

%e For T(3,1) solution is |{(3)}| = 1.

%e For T(3,2) solution is |{(1,3), (2,3), (3,3)}| = 3.

%e 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.

%t T[n_, k_] := Which[

%t   k == 1, 1,

%t   k == n, Binomial[2n-1, n],

%t   k == n-1, T[n-1, n-1],

%t   1 < k < n-1, T[n-1, k] + T[n, k-1]

%t ];

%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten

%o (PARI) T(n,k)={if(k<n, binomial(k+n-2,n-1), binomial(2*n-1,n))} \\ _Andrew Howroyd_, Nov 24 2024

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

%K nonn,tabl,new

%O 1,3

%A _Zlatko Damijanic_, Nov 24 2024