OEIS A000081: Rooted Trees and Hidden Connections

A000081 counts unlabeled rooted trees with n nodes, but it surfaces in several surprising guises: arrangements of non-overlapping circles, connected endofunctions with fixed points, and connected multigraphs with a single loop and no other cycles. We’ll unpack the elegant generating function T(x) = x · exp( sum_{k≥1} T(x^k)/k ), illustrate how Lagrange inversion yields the coefficients, and discuss Otter’s asymptotics a(n) ~ C · ρ^n · n^{-3/2} with ρ ≈ 2.955765 and C ≈ 0.5349. You’ll hear how the same counting rules (via cycle indices of permutations) give all these interpretations, and we’ll set the record straight on the early terms (1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, …). We’ll also glimpse the circle-picture and the one-loop multigraph connections that reveal why this single sequence sits at the crossroads of combinatorics, graphs, and functional digraphs. Join us as we pull back the curtain on the hidden structure behind A000081.

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