OEIS A000307: Four-level labeled rooted trees

We explore A000307, the count of four-level labeled rooted trees with n leaves. Despite the OEIS tag 'easy', the numbers explode quickly, and the mystery is why this structure is considered easy in theory. We explain what 'easy' means in OEIS terms, how the nested generating function exp(exp(exp(exp(x)))-1) encodes the four-level construction, and present the alternative Stirling-number sum formula that offers a different combinatorial viewpoint. Along the way we’ll note the initial terms (1, 1, 4, 22, 154) and discuss why this elegant object sits at the intersection of recursion, generating functions, and partitions.

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