%I A001699 M3087 N1251
%S A001699 1,1,3,21,651,457653,210065930571,44127887745696109598901,
%T A001699 1947270476915296449559659317606103024276803403,
%U A001699 3791862310265926082868235028027893277370233150300118107846437701158064808916492244872560821
%N A001699 Number of binary trees of height n; or products (ways to insert parentheses) 
               of height n when multiplication is non-commutative and non-associative.
%C A001699 Approaches 1.5028368...^(2^n). Row sums of A065329 as square array. - 
               Henry Bottomley (se16(AT)btinternet.com), Oct 29 2001. Also row sum 
               of square array A073345 (AK).
%D A001699 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001699 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001699 I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. 
               Edinburgh, 59 (Part 2, 1938-39), 153-162.
%D A001699 T. K. Moon, Enumerations of binary trees, types of trees and the number 
               of reversiblevariable length codes, submitted to Discrete Applied 
               Mathematics, 2000.
%H A001699 David Wasserman, <a href="b001699.txt">Table of n, a(n) for n = 0..12</
               a> [Shortened file because terms grow rapidly: see Wasserman link 
               below for an additional term]
%H A001699 A. V. Aho and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               doc/doubly.html">Some doubly exponential sequences</a>, Fib. Quart., 
               11 (1973), 429-437.
%H A001699 H. Bottomley, <a href="a001699.gif">Illustration of initial terms</a>
%H A001699 C. Lenormand, <a href="http://www.ai.univ-paris8.fr/~lenormand/I.2_Magmas_Arborescences.pdf">
               Arbres et permutations II</a>, see p. 6
%H A001699 David Wasserman, <a href="a001699.txt">Table of n, a(n) for n = 0..13</
               a>
%H A001699 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BinaryTree.html">Link to a section of The World of Mathematics.</
               a>
%H A001699 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A001699 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%H A001699 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%H A001699 <a href="Sindx_Par.html#parens">Index entries for sequences related to 
               parenthesizing</a>
%H A001699 <a href="Sindx_Aa.html#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 
               + ...</a>
%F A001699 a(n+1) = 2*a(n)*(a(0)+...+a(n-1))+a(n)^2.
%F A001699 a(n+1) = a(n)^2+a(n)+a(n)*sqrt(4*a(n)-3), if n>0.
%F A001699 a(n+1) = A003095(n+1)-A003095(n) = A003095(n)^2- A003095(n)+1. - Henry 
               Bottomley (se16(AT)btinternet.com), Apr 26 2001
%F A001699 a(n)=A059826(A003095(n-1))
%p A001699 s := proc(n) local i,j,ans; ans := [ 1 ]; for i to n do ans := [ op(ans),
               2*(add(j,j=ans)-ans[ i ])*ans[ i ]+ans[ i ]^2 ] od; RETURN(ans); 
               end; s(10);
%o A001699 (PARI) a(n)=if(n<=1,n >= 0,a(n-1)*(a(n-1)+a(n-2)+a(n-1)/a(n-2))); b(n)=if(n<1,
               0,1+b(n-1)^2); A003095(n)=b(n); A059826(n)=(n^2-n+1)*(n^2+n+1); A002061(n)=n^2-n+1
%Y A001699 Cf. A002658, A056207, A002449, A003095.
%Y A001699 Cf. A004019.
%Y A001699 Sequence in context: A093549 A012044 A098918 this_sequence A162924 A057600 
               A079269
%Y A001699 Adjacent sequences: A001696 A001697 A001698 this_sequence A001700 A001701 
               A001702
%K A001699 nonn,easy,core,nice
%O A001699 0,3
%A A001699 N. J. A. Sloane (njas(AT)research.att.com), Jeffrey Shallit