OEIS A000242: Third power of rooted tree enumerator; number of linear forests of three rooted trees

We explore OEIS A000242, which is the cube of the rooted-tree enumerator. Its generating function is B(x)^3, where B(x) is the generating function for rooted trees (A000081). Thus A000242 counts ordered triples of rooted trees with a total of n nodes, i.e., linear forests of three rooted trees. The offset is 3, so the sequence starts at n=3 with A3=1, A4=3, A5=9, etc. This ties the combinatorics of rooted trees to threefold convolutions of their counts. The page also notes historical identifiers M2798 and L126.

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