OEIS A000269: Three Labeled Vertices in Trees

In this episode we unpack A000269—the count of trees on n nodes with three distinct vertices labeled. The first terms are 3 for n=3 and 16 for n=4, with numbers growing rapidly as n increases. The entry carries the nice/easy labels because the underlying structure is surprisingly elegant: a generating-function relation ties A000269 to the rooted-tree generating function A000081, and there’s also a straightforward arithmetic form A(n) = a00024(n) − 2·a000243. These formulas show how counting with a small labeling constraint reduces to rooted-tree decompositions and inclusion–exclusion ideas. The sequence has a storied pedigree (Riordan, Sloan) and serves as a clear teaching example of using generating functions to relate labeled and rooted structures. If you’re a number-theory student, reflect on how marking three vertices reshapes the counting landscape and what that reveals about structure and labeling.

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