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Triangle, read by rows of terms T(n,k) for k=0..n^2, that starts with a '1' in row 0 with row n>0 consisting of 2*n-1 '1's followed by the partial sums of the prior row.
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%I #19 Jan 15 2025 11:48:09

%S 1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,3,4,6,1,1,1,1,1,1,1,1,2,3,4,5,6,8,11,

%T 15,21,1,1,1,1,1,1,1,1,1,1,2,3,4,5,6,7,8,10,13,17,22,28,36,47,62,83,1,

%U 1,1,1,1,1,1,1,1,1,1,1,2,3,4,5,6,7,8,9,10,12,15,19,24,30,37,45,55,68,85,107,135,171,218,280,363

%N Triangle, read by rows of terms T(n,k) for k=0..n^2, that starts with a '1' in row 0 with row n>0 consisting of 2*n-1 '1's followed by the partial sums of the prior row.

%C Right border and row sums form A178325.

%H Paul D. Hanna, <a href="/A214403/b214403.txt">Rows n = 0..14, flattened.</a>

%e Triangle begins:

%e [1];

%e [1, 1];

%e [1,1,1, 1, 2];

%e [1,1,1,1,1, 1,2,3, 4, 6];

%e [1,1,1,1,1,1,1, 1,2,3,4,5, 6,8,11, 15, 21];

%e [1,1,1,1,1,1,1,1,1, 1,2,3,4,5,6,7, 8,10,13,17,22, 28,36,47, 62, 83];

%e ...

%e Row sums equal the row sums (A178325) of triangle A214398,

%e where A214398(n, k) = binomial(k^2+n-k-1, n-k):

%e 1;

%e 1, 1;

%e 1, 4, 1;

%e 1, 10, 9, 1;

%e 1, 20, 45, 16, 1;

%e 1, 35, 165, 136, 25, 1;

%e 1, 56, 495, 816, 325, 36, 1;

%e 1, 84, 1287, 3876, 2925, 666, 49, 1;

%e ...

%o (PARI) {T(n, k)=if(k>n^2||n<0||k<0, 0, if(n==0,1,if(k<=2*n-1, 1, sum(i=0, k-2*n+1, T(n-1, i)))))}

%o for(n=0,10,for(k=0,n^2,print1(T(n,k),", "));print(""))

%Y Cf. A131338, A178325, A214398.

%K nonn,tabf

%O 0,8

%A _Paul D. Hanna_, Jul 15 2012