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.
The number of terms in row n is C(n,[n/2]). Triangle begins: [1], [2], [3,1], [4,2,2], [5,3,3,3,1,1], [6,4,4,4,4,2,2,2,2,2], [7,5,5,5,5,5,3,3,3,3,3,3,3,3,3,1,1,1,1,1], [8,6,6,6,6,6,6,4,4,4,4,4,4,4,4,4,4,4,4,4,4,2,2,2,2,2,2,2,2,2,2,2,2,2,2], [9,7,7,7,7,7,7,7,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1], ... ILLUSTRATION OF GENERATING METHOD. Row n is derived from the binomial coefficients in the following way. Place markers in an array so that the number of contiguous markers in row k is C(n,k) and then count the markers along columns. For example, row 6 of this triangle is generated from C(6,k) like so: ------------------------------------------ 1: o - - - - - - - - - - - - - - - - - - - 6: o o o o o o - - - - - - - - - - - - - - 15:o o o o o o o o o o o o o o o - - - - - 20:o o o o o o o o o o o o o o o o o o o o 15:o o o o o o o o o o o o o o o - - - - - 6: o o o o o o - - - - - - - - - - - - - - 1: o - - - - - - - - - - - - - - - - - - - ------------------------------------------ Counting the markers along the columns gives row 6 of this triangle: [7,5,5,5,5,5,3,3,3,3,3,3,3,3,3,1,1,1,1,1]. Continuing in this way generates all the rows of this triangle. ... Number of repeated terms in each row of this triangle forms A008315: 1; 1; 1, 1; 1, 2; 1, 3, 2; 1, 4, 5; 1, 5, 9, 5; 1, 6, 14, 14; 1, 7, 20, 28, 14;...
{T(n,k)=polcoeff(sum(j=0,n,(x^binomial(n,j) - 1)/(x-1)),k)}
for(n=0,10, for(k=0, binomial(n,n\2)-1, print1(T(n,k),","));print(""))