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 A294198 Labeled trees on n nodes with at least one node of degree two. 1
 0, 0, 3, 12, 120, 1200, 16380, 255696, 4726008, 99107280, 2346042600, 61706210280, 1788467429892, 56618211155688, 1944581982268380, 72019509651227040, 2861473883255362800, 121414006972684901664, 5479661140682410928592, 262122044885503316203320 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Table of n, a(n) for n=1..20. Marko Riedel, Maple code for sequence including total enumeration and closed form. Marko Riedel et al., How many trees have a vertex of degree two FORMULA a(n) = (n-2)!*Sum_{q=1..n-2} C(n,q)*(-1)^(q+1)*(n-q)^(n-2-q)/(n-2-q)! where n >= 2. EXAMPLE When n=4 there are two types of trees: paths (24!/2 trees) or a star (4 trees) for a total of 4^(4-2) trees. Of these only the paths contain a vertex of degree 2 (2 in fact). When n=5 there are three types of trees: paths (5!/2 trees), a star (5 trees) or a fork (C(5,1) x C(4, 2) x 2) for a total of 5^(5-2) trees. Of these only the paths and the fork contain a vertex of degree 2. MATHEMATICA a[1] = 0; a[n_] := (n-2)! Sum[Binomial[n, q] (-1)^(q+1) (n-q)^(n-2-q) / (n-2-q)!, {q, 1, n-2}]; Array[a, 20] (* Jean-François Alcover, Feb 15 2018 *) CROSSREFS Cf. A007106, A274699. Sequence in context: A359658 A194506 A280458 * A133553 A010571 A280248 Adjacent sequences: A294195 A294196 A294197 * A294199 A294200 A294201 KEYWORD nonn AUTHOR Marko Riedel, Feb 11 2018 STATUS approved

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Last modified June 15 11:51 EDT 2024. Contains 373407 sequences. (Running on oeis4.)