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%I #12 Jun 14 2017 00:31:48
%S 1,1,1,1,1,1,2,1,1,1,1,2,3,5,1,1,1,1,1,2,3,4,6,9,14,1,1,1,1,1,1,2,3,4,
%T 5,7,10,14,20,29,43,1,1,1,1,1,1,1,2,3,4,5,6,8,11,15,20,27,37,51,71,
%U 100,143,1,1,1,1,1,1,1,1,2,3,4,5,6,7,9,12,16,21,27,35,46,61,81,108,145,196
%N Triangle, read by rows of n*(n+1)/2 + 1 terms, that starts with a '1' in row 0 with row n consisting of n '1's followed by the partial sums of the prior row.
%H Paul D. Hanna, <a href="/A131338/b131338.txt">Rows n = 0..16, flattened.</a>
%F T(n,k) = Sum_{i=0..k-n} T(n-1,i) for k>n, else T(n,k)=1 for n>=k>=0.
%F Right border: T(n+1, (n+1)*(n+2)/2) = A098569(n) = Sum_{k=0..n} C( (k+1)*(k+2)/2 + n-k-1, n-k).
%F T(n, n*(n-1)/2 + 1) = Sum_{k=0..n-1} C(k*(k+1)/2, n-k) = A121690(n-1) for n>=1. - _Paul D. Hanna_, Aug 30 2007
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1,1, 1,2;
%e 1,1,1, 1,2,3,5;
%e 1,1,1,1, 1,2,3,4,6,9,14;
%e 1,1,1,1,1, 1,2,3,4,5,7,10,14,20,29,43;
%e 1,1,1,1,1,1, 1,2,3,4,5,6,8,11,15,20,27,37,51,71,100,143;
%e 1,1,1,1,1,1,1, 1,2,3,4,5,6,7,9,12,16,21,27,35,46,61,81,108,145,196,267,367,510; ...
%e Row sums equal the row sums (A098569) of triangle A098568,
%e where A098568(n, k) = binomial( (k+1)*(k+2)/2 + n-k-1, n-k):
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 6, 6, 1;
%e 1, 10, 21, 10, 1;
%e 1, 15, 56, 55, 15, 1;
%e 1, 21, 126, 220, 120, 21, 1; ...
%o (PARI) T(n,k)=if(k>n*(n+1)/2 || k<0,0,if(k<=n,1,sum(i=0,k-n,T(n-1,i))))
%o for(n=0, 10, for(k=0, n*(n+1)/2, print1(T(n, k), ", ")); print(""))
%Y Cf. A098568, A098569 (row sums), A121690, A183202.
%Y Cf. A214403 (variant).
%K nonn,tabl
%O 0,7
%A _Paul D. Hanna_, Jun 29 2007
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