Form a conjugate partition of row with 1+1+1 in first row. all other rows are the union of their parents. n-th row sum is equal to 3*2^(n-1). The largest part of n-th row is A000204(n). a(n) = number of types of piles in n-th row.

(first row: 1+1+1 and the other conjugate partition is 3; 2nd row is union of 1+1+1 and 3.) (2nd row: 3+1+1+1 and the other conjugate partition is 4+1+1; 3rd row is union of 3+1+1+1 and 4+1+1.) (3rd row: 4+3+1+1+1+1+1 and the other conjugate partition is 7+2+2+1; 4th row is union of 4+3+1+1+1+1+1 and 7+2+2+1.)