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 A035010 Number of prime binary rooted trees with n external nodes. 3
 1, 2, 4, 14, 38, 132, 420, 1426, 4834, 16796, 58688, 208012, 742636, 2674384, 9693976, 35357670, 129641774, 477638700, 1767253368, 6564119892, 24466233428, 91482563640, 343059494120, 1289904147128, 4861945985428, 18367353066440, 69533549429280, 263747951750360 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 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 nodes. 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. REFERENCES B. Amerlynck, Itérées d'exponentielles: aspects combinatoires et arithmétiques, Mémoire de licence, Univ. Libre de Bruxelles (1998). LINKS Alois P. Heinz, Table of n, a(n) for n = 2..1000 V. Blondel, Une famille d'opérations sur les arbres binaires, [A family of operations on binary trees], Comptes Rendus de l'Academie des Sciences de Paris - Serie I, 321, 491-494, 1995. V. Blondel, Structured numbers: properties of a hierarchy of operations on binary trees, Acta Informatica, vol. 35 (1998), pp. 1-15. FORMULA a(n) = C(n-1) - sum_{d_1*d_2=n and 1 binomial(2*n, n)/(n+1): a:= proc(n) option remember; C(n-1)       -add(a(d)*C(n/d-1), d=divisors(n) minus {1, n})     end: seq(a(n), n=2..30);  # Alois P. Heinz, Feb 12 2015 MATHEMATICA a[n_] := a[n] = CatalanNumber[n-1] - Sum[If[Divisible[n, d1], d2 = n/d1; a[d1]*CatalanNumber[d2-1], 0], {d1, 2, n-1}]; a[2] = 1; Table[a[n], {n, 2, 26}] (* Jean-François Alcover, Oct 25 2011, after formula *) CROSSREFS Cf. A035102. Sequence in context: A006611 A007462 A053623 * A055540 A006252 A079995 Adjacent sequences:  A035007 A035008 A035009 * A035011 A035012 A035013 KEYWORD nice,easy,nonn AUTHOR Bernard Amerlynck (B.Amerlynck(AT)ulg.ac.be) EXTENSIONS More terms from Christian G. Bower STATUS approved

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Last modified August 18 15:00 EDT 2019. Contains 326106 sequences. (Running on oeis4.)