Number of irreducible partitions into smaller triangular numbers. A partition is irreducible if no subpartition with 2 or more parts sums to a smaller triangular number.

a(9)=1 for the partition [6,3]. [6,1^3], [3^3], [3^2,1^3], [3,1^6] and [1^9] are all excluded because they contain subpartitions [3^2] or [1^3] summing to a triangular number.