OEIS A00299: Number of n-node rooted trees of height 4

We dive into OEIS A00299—the count of n-node rooted trees of height 4. We’ll unpack why fixing height sparks explosive growth (zeros for n<5, then 1, 4, 13, 36, and beyond, eventually billions by n=34), and what this says about how simple rules generate complex structures. We’ll place the sequence in its historical context—from the OEIS’s development and Harary–Robinson–Riordan methods to modern computations—and reflect on what other simple parameters might hide vast counting landscapes waiting to be discovered.

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