1,3

2nd column of triangular array A035342 whose first column is given by A001147(n), n >= 1. Recursion: a(n) = 2*n*a(n-1)+ A001147(n-1), n >= 2, a(1)=0.

a(n) gives the number of organically labeled forests (sets) with two rooted ordered trees with n non-root vertices. See the example a(3)=9 given in A035342. Organic labelling means that the vertex labels along the (unique) path from the root to any of the leaves (degree 1, non-root vertices) is increasing. W. Lang, Aug 07 2007.

a(n), n>=2, enumerates unordered n-vertex forests composed of two plane (ordered) ternary (3-ary) trees with increasing vertex labeling. See A001147 (number of increasing ternary trees) and a D. Callan comment there. For a picture of some ternary trees see a W. Lang link under A001764.

a(n) = n!*A008549(n-1)/2^(n-1)= n!(4^(n-1)-binomial(2*n, n)/2)/2^(n-1).

a(2)=1 for the forest: {r1-1, r2-2} (with root labels r1 and r2). The order between the components of the forest is irrelevant (like for sets).

a(3)=9 increasing ternary 2-forest with n=3 vertices: there are three 2-forests (the one vertex tree together with any of the three different 2-vertex trees) each with three increasing labelings. W. Lang, Sep 14 2007.

Related to A008549 and A001147.

Cf. A000108, A035342.

Cf. A001147 (m=1 column of A035342). See a D. Callan comment there on the number of increasing ordered rooted trees on n+1 vertices.

Sequence in context: A028339 A100814 A055725 this_sequence A015583 A084022 A084015

Adjacent sequences: A035098 A035099 A035100 this_sequence A035102 A035103 A035104

easy,nonn

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)