OEIS A001168: Rooted planar maps

We explore A001168, the counting sequence for rooted planar maps with n edges. The nth term is A_n = 2 · 3^n · (2n)! / (n! (n+2)!). We unpack why this simple closed form hints at deep structure, its duality connection to rooted 4-regular planar maps, and the surprising web of related objects (doodles, lambda calculus, well-labeled trees, and the Tamari lattice) highlighted by Noam Zeilberger. We also touch on the integral representation as the nth moment of a positive function and what the asymptotics tell us about the growth and underlying analytic structure of these maps.

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