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.
Triangle begins: 1; 0, 1; 0, 2, 4; 0, 4, 12, 18; 0, 8, 32, 70, 94; 0, 16, 80, 224, 426, 544; ... T(4,3) = 70 because it is the sum of the entry to the left (32) plus the sum of all the entries above position (4,3), which give 1 + 0 + 1 + 2 + 4 + 12 + 18.
T := proc(i,j) option remember; local r,s,t1; if i=0 and j=0 then RETURN(1); fi; if j=0 then RETURN(0); fi; t1 := T(i,j-1); for r from 0 to i-j do for s from 0 to j do if r+s <> i then t1 := t1+T(r+s,s); fi; od: od: RETURN(t1); end; # n-th row is T(n,0), T(n,1), ..., T(n,n) To get the triangle formed when the left diagonal has a single 1 in position k: T := proc(i,j,k) option remember; local r,s,t1; if i < k then RETURN(0); fi; if i = k then RETURN(1); fi; if j = 0 then RETURN(0); fi; t1 := T(i,j-1,k); for r from 0 to i-j do for s from 0 to j do if r+s <> i then t1 := t1+T(r+s,s,k); fi; od: od: t1; end;
T [i_, j_] := T[i, j] = Module[{r, s, t1}, If[i == 0 && j == 0, Return[1]]; If[j == 0, Return[0]]; t1 = T[i, j-1]; For[r = 0, r <= i-j, r++, For[s = 0, s <= j, s++, If[r+s != i, t1 = t1 + T[r+s, s]]]]; Return[t1]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 26 2013, translated from Maple *)
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