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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.

%I A378377 #19 Dec 06 2024 11:18:29
%S A378377 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 A378377 462,1716,1,8,36,120,330,792,1716,6435,1,9,45,165,495,1287,3003,6435,
%U A378377 24310,1,10,55,220,715,2002,5005,11440,24310,92378
%N 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.
%C A378377 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 A378377 When k < n, then the last number is n.
%F A378377 T(n,n) = binomial(2*n-1,n).
%F A378377 T(n,k) = binomial(k+n-2, n-1) for k < n.
%e A378377 Triangle begins:
%e A378377   1
%e A378377   1 3
%e A378377   1 3 10
%e A378377   1 4 10 35
%e A378377   1 5 15 35 126
%e A378377   1 6 21 56 126 462
%e A378377   1 7 28 84 210 462 1716
%e A378377   ...
%e A378377 For T(3,1) solution is |{(3)}| = 1.
%e A378377 For T(3,2) solution is |{(1,3), (2,3), (3,3)}| = 3.
%e A378377 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 A378377 T[n_, k_] := Which[
%t A378377   k == 1, 1,
%t A378377   k == n, Binomial[2n-1, n],
%t A378377   k == n-1, T[n-1, n-1],
%t A378377   1 < k < n-1, T[n-1, k] + T[n, k-1]
%t A378377 ];
%t A378377 Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten
%o A378377 (PARI) T(n,k)={if(k<n, binomial(k+n-2,n-1), binomial(2*n-1,n))} \\ _Andrew Howroyd_, Nov 24 2024
%Y A378377 Cf. A051924 (row sums), A001700 (right diagonal).
%K A378377 nonn,tabl
%O A378377 1,3
%A A378377 _Zlatko Damijanic_, Nov 24 2024