OEIS A000243: Trees with two labeled nodes

Explore A000243, the count of unrooted trees on n nodes with two distinct labeled vertices. We’ll cover the offset (n=2 corresponds to 1) and the early terms (1, 3, 9, 26, 75, 214, …), its connections to rooted trees and the broader A034799 table, and the multiple formulas and generating-function viewpoints that explain how these numbers are built. We’ll also touch on the historical context—from Sloan to Riordan—and note the surprising crosslinks, such as a match with counts of non-intersecting circles in the plane, illustrating how a single sequence sits at the crossroads of different combinatorial ideas.

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