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A115323 Triangle, read by rows, where row n+1 is formed by sorting, in ascending order, the result of the convolution of row n (consisting of 2n+1 terms) with [1,1,1].

%I A115323 #5 Jun 13 2017 23:40:49
%S A115323 1,1,1,1,1,1,2,2,3,1,2,3,4,5,5,7,1,3,6,7,9,12,12,14,17,1,4,10,16,17,
%T A115323 22,28,31,33,38,43,1,5,15,30,43,43,55,67,81,81,92,102,114,1,6,21,50,
%U A115323 88,114,116,141,165,203,216,229,254,275,308,1,7,28,77,159,252,308,318,371,422
%N A115323 Triangle, read by rows, where row n+1 is formed by sorting, in ascending order, the result of the convolution of row n (consisting of 2n+1 terms) with [1,1,1].
%e A115323 Convolution of [1,1,1] with row 3 gives:
%e A115323 [1,1,1]*[1,2,3,4,5,5,7] = [1,3,6,9,12,14,17,12,7];
%e A115323 when sorted yields row 4: [1,3,6,7,9,12,12,14,17].
%e A115323 Triangle begins:
%e A115323 1;
%e A115323 1,1,1;
%e A115323 1,1,2,2,3;
%e A115323 1,2,3,4,5,5,7;
%e A115323 1,3,6,7,9,12,12,14,17;
%e A115323 1,4,10,16,17,22,28,31,33,38,43;
%e A115323 1,5,15,30,43,43,55,67,81,81,92,102,114;
%e A115323 1,6,21,50,88,114,116,141,165,203,216,229,254,275,308;
%e A115323 1,7,28,77,159,252,308,318,371,422,509,583,584,648,699,758,837; ...
%o A115323 (PARI) T(n,k)=local(V);if(2*n<k || k<0,0,if(k==0 || n==1,1,V=vector(2*n+1); for(i=0,2*n,V[i+1]=if(i-2>=0,T(n-1,i-2))+if(i>0,T(n-1,i-1))+T(n-1,i)); V=vecsort(V);V[k+1]))
%Y A115323 Cf. A103284 (variant), A115324 (central terms), A115325 (diagonal).
%K A115323 nonn,tabl
%O A115323 0,7
%A A115323 _Paul D. Hanna_, Jan 20 2006