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A183202 Triangle, read by rows, where T(n,k) equals the sum of (n-k) terms in row n of triangle A131338 starting at position nk - k(k-1)/2, with the main diagonal formed from the row sums.

%I A183202 #10 Mar 29 2025 18:16:03
%S A183202 1,1,1,2,1,2,3,3,3,5,4,6,10,9,14,5,10,22,34,29,43,6,15,40,84,122,100,
%T A183202 143,7,21,65,169,334,463,367,510,8,28,98,300,738,1390,1851,1426,1936,
%U A183202 9,36,140,489,1426,3345,6043,7767,5839,7775,10,45,192,749,2510,6990,15735
%N A183202 Triangle, read by rows, where T(n,k) equals the sum of (n-k) terms in row n of triangle A131338 starting at position nk - k(k-1)/2, with the main diagonal formed from the row sums.
%e A183202 Triangle begins:
%e A183202 1;
%e A183202 1,1;
%e A183202 2,1,2;
%e A183202 3,3,3,5;
%e A183202 4,6,10,9,14;
%e A183202 5,10,22,34,29,43;
%e A183202 6,15,40,84,122,100,143;
%e A183202 7,21,65,169,334,463,367,510;
%e A183202 8,28,98,300,738,1390,1851,1426,1936;
%e A183202 9,36,140,489,1426,3345,6043,7767,5839,7775;
%e A183202 10,45,192,749,2510,6990,15735,27374,34097,25094,32869; ...
%e A183202 The rows are derived from triangle A131338 by summing terms in the following manner:
%e A183202 (1);
%e A183202 (1),(1);
%e A183202 (1+1),(1),(2);
%e A183202 (1+1+1),(1+2),(3),(5);
%e A183202 (1+1+1+1),(1+2+3),(4+6),(9),(14);
%e A183202 (1+1+1+1+1),(1+2+3+4),(5+7+10),(14+20),(29),(43);
%e A183202 (1+1+1+1+1+1),(1+2+3+4+5),(6+8+11+15),(20+27+37),(51+71),(100),(143); ...
%e A183202 where row n of triangle A131338 consists of n '1's followed by the partial sums of the prior row.
%o A183202 (PARI) {A131338(n, k)=if(k>n*(n+1)/2||k<0,0,if(k<=n,1,sum(i=0, k-n,A131338(n-1,i))))}
%o A183202 {T(n,k)=if(n==k,A131338(n,n*(n+1)/2),sum(j=n*k-k*(k-1)/2,n*k-k*(k-1)/2+n-k-1,A131338(n,j)))}
%Y A183202 Cf. A131338, A098568, A098569 (row sums), A183203 (antidiagonal sums).
%K A183202 tabl,nonn
%O A183202 0,4
%A A183202 _Paul D. Hanna_, Dec 30 2010