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A214403 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.

%I A214403 #19 Jan 15 2025 11:48:09
%S A214403 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 A214403 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 A214403 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 A214403 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 A214403 Right border and row sums form A178325.
%H A214403 Paul D. Hanna, <a href="/A214403/b214403.txt">Rows n = 0..14, flattened.</a>
%e A214403 Triangle begins:
%e A214403   [1];
%e A214403   [1, 1];
%e A214403   [1,1,1, 1, 2];
%e A214403   [1,1,1,1,1, 1,2,3, 4, 6];
%e A214403   [1,1,1,1,1,1,1, 1,2,3,4,5, 6,8,11, 15, 21];
%e A214403   [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 A214403   ...
%e A214403 Row sums equal the row sums (A178325) of triangle A214398,
%e A214403 where A214398(n, k) = binomial(k^2+n-k-1, n-k):
%e A214403   1;
%e A214403   1, 1;
%e A214403   1, 4, 1;
%e A214403   1, 10, 9, 1;
%e A214403   1, 20, 45, 16, 1;
%e A214403   1, 35, 165, 136, 25, 1;
%e A214403   1, 56, 495, 816, 325, 36, 1;
%e A214403   1, 84, 1287, 3876, 2925, 666, 49, 1;
%e A214403   ...
%o A214403 (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 A214403 for(n=0,10,for(k=0,n^2,print1(T(n,k),", "));print(""))
%Y A214403 Cf. A131338, A178325, A214398.
%K A214403 nonn,tabf
%O A214403 0,8
%A A214403 _Paul D. Hanna_, Jul 15 2012