%I A131338
%S A131338 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,
               5,
%T A131338 7,10,14,20,29,43,1,1,1,1,1,1,1,2,3,4,5,6,8,11,15,20,27,37,51,71,100,
%U A131338 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 A131338 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.
%F A131338 T(n,k) = Sum_{i=0..k-n} T(n-1,i) for k>n, else T(n,k)=1 for n>=k>=0. 
               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 A131338 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 (pauldhanna(AT)juno.com), Aug 30 2007
%e A131338 Triangle begins:
%e A131338 1;
%e A131338 1, 1;
%e A131338 1,1, 1,2;
%e A131338 1,1,1, 1,2,3,5;
%e A131338 1,1,1,1, 1,2,3,4,6,9,14;
%e A131338 1,1,1,1,1, 1,2,3,4,5,7,10,14,20,29,43;
%e A131338 1,1,1,1,1,1, 1,2,3,4,5,6,8,11,15,20,27,37,51,71,100,143;
%e A131338 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 A131338 Row sums equal the row sums (A098569) of triangle A098568,
%e A131338 where A098568(n, k) = C( (k+1)*(k+2)/2 + n-k-1, n-k):
%e A131338 1;
%e A131338 1,1;
%e A131338 1,3,1;
%e A131338 1,6,6,1;
%e A131338 1,10,21,10,1;
%e A131338 1,15,56,55,15,1;
%e A131338 1,21,126,220,120,21,1; ...
%o A131338 (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))))}
%Y A131338 Cf. A098568, A098569 (row sums).
%Y A131338 Cf. A121690.
%Y A131338 Sequence in context: A029384 A094102 A063746 this_sequence A106498 A093466 
               A125761
%Y A131338 Adjacent sequences: A131335 A131336 A131337 this_sequence A131339 A131340 
               A131341
%K A131338 nonn,tabl
%O A131338 0,7
%A A131338 Paul D. Hanna (pauldhanna(AT)juno.com), Jun 29 2007