A035102 Composite binary rooted trees with external nodes.

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

Offset: 2

Author

Bernard AMERLYNCK (B.Amerlynck(AT)ulg.ac.be)

Keywords

• nonn

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

• Index entries for sequences related to rooted trees

Crossrefs

Cf. A035010.

Programs

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 *)

Formula

A035010(n)+A035102(n)=Catalan(n-1)=A000108(n-1).