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 A035102 Composite binary rooted trees with external nodes. 3
 0, 0, 1, 0, 4, 0, 9, 4, 28, 0, 98, 0, 264, 56, 869, 0, 3016, 0, 9822, 528, 33592, 0, 119530, 196, 416024, 5712, 1486724, 0, 5369336, 0, 19392637, 67184, 70715340, 3696, 259535958, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,5 COMMENTS If a,b are binary trees, a.b is equal to tree b where a copy of a is put on each of b's external node. This is non-commutative but associative. A binary tree a is prime if it is different from the 1 node tree and if a=b.c implies that b or c is equal to the 1 node tree. LINKS FORMULA A035010(n)+A035102(n)=Catalan(n-1)=A000108(n-1). MATHEMATICA (* b = A035010 *) b[n_] := b[n] = CatalanNumber[n-1] - Sum[If[Divisible[n, d1], d2 = n/d1; b[d1]*CatalanNumber[d2-1], 0], {d1, 2, n-1}]; b[2] = 1; a[n_] := a[n] = CatalanNumber[n-1] - b[n]; Table[a[n], {n, 2, 37}] (* Jean-François Alcover, Jul 17 2012, after formula *) CROSSREFS Cf. A035010. Sequence in context: A338107 A100074 A330422 * A242015 A187507 A187857 Adjacent sequences:  A035099 A035100 A035101 * A035103 A035104 A035105 KEYWORD nonn AUTHOR Bernard AMERLYNCK (B.Amerlynck(AT)ulg.ac.be) STATUS approved

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Last modified December 3 12:41 EST 2021. Contains 349463 sequences. (Running on oeis4.)