This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A128320 #12 Jul 03 2024 19:26:00
%S A128320 1,1,1,4,3,1,17,8,5,1,98,41,12,7,1,622,234,73,16,9,1,4512,1602,418,
%T A128320 113,20,11,1,35373,11976,3110,650,161,24,13,1,300974,98541,23920,5242,
%U A128320 930,217,28,15,1,2722070,866942,207549,41304,8094,1258,281,32,17,1
%N A128320 Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n that are to the right of T(n,k) with the vector of terms in column k that are above T(n,k) for n>k+1>0, with the odd numbers in the secondary diagonal and all 1's in the main diagonal.
%H A128320 G. C. Greubel, <a href="/A128320/b128320.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A128320 T(n,k) = Sum_{j=0..n-1-k} T(n,k+j+1)*T(k+j,k) for n > k+1 > 0, with T(n,n) = 1 and T(n, n-1) = 2*n-1 for k >= 0.
%e A128320 Illustrate the recurrence by:
%e A128320 T(n,k) = [T(n,k+1),T(n,k+2), ..,T(n,n)]*[T(k,k),T(k+1,k),..,T(n-1,k)]:
%e A128320 T(3,0) = [8,5,1]*[1,1,4]~ = 8*1 + 5*1 + 1*4 = 17;
%e A128320 T(4,1) = [12,7,1]*[1,3,8]~ = 12*1 + 7*3 + 1*8 = 41;
%e A128320 T(5,1) = [73,16,9,1]*[1,3,8,41]~ = 73*1 + 16*3 + 9*8 + 1*41 = 234;
%e A128320 T(6,2) = [113,20,11,1]*[1,5,12,73]~ = 113*1 + 20*5 + 11*12 + 1*73 = 418.
%e A128320 Triangle begins:
%e A128320 1;
%e A128320 1, 1;
%e A128320 4, 3, 1;
%e A128320 17, 8, 5, 1;
%e A128320 98, 41, 12, 7, 1;
%e A128320 622, 234, 73, 16, 9, 1;
%e A128320 4512, 1602, 418, 113, 20, 11, 1;
%e A128320 35373, 11976, 3110, 650, 161, 24, 13, 1;
%e A128320 300974, 98541, 23920, 5242, 930, 217, 28, 15, 1;
%e A128320 2722070, 866942, 207549, 41304, 8094, 1258, 281, 32, 17, 1;
%e A128320 26118056, 8139602, 1885166, 377757, 65088, 11762, 1634, 353, 36, 19, 1;
%t A128320 T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==n-1, 2*n-1, Sum[T[n,k+j+1] *T[k+j,k], {j,0,n-k-1}]]];
%t A128320 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 25 2024 *)
%o A128320 (PARI)
%o A128320 {T(n,k)=if(n==k,1, if(n==k+1,2*n-1, sum(i=0,n-k-1, T(n,k+i+1)*T(k+i,k))))};
%o A128320 for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
%o A128320 (Magma)
%o A128320 function T(n,k) // T = A128320
%o A128320 if k eq n then return 1;
%o A128320 elif k eq n-1 then return 2*n-1;
%o A128320 else return (&+[T(n, k+j+1)*T(k+j, k): j in [0..n-k-1]]);
%o A128320 end if;
%o A128320 end function;
%o A128320 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 25 2024
%o A128320 (SageMath)
%o A128320 @CachedFunction
%o A128320 def T(n,k): # T = A128320
%o A128320 if k==n: return 1
%o A128320 elif k==n-1: return 2*n-1
%o A128320 else: return sum(T(n, k+j+1)*T(k+j, k) for j in range(n-k))
%o A128320 flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jun 25 2024
%Y A128320 Columns k: A128321 (k=0), A128322 (k=1), A128323 (k=2).
%Y A128320 Sums: A128324 (row sums).
%Y A128320 Variant of: A115080.
%K A128320 nonn,tabl
%O A128320 0,4
%A A128320 _Paul D. Hanna_, Feb 25 2007