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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.
(Formerly M3087 N1251)

%I M3087 N1251

%S 1,1,3,21,651,457653,210065930571,44127887745696109598901,

%T 1947270476915296449559659317606103024276803403,

%U 3791862310265926082868235028027893277370233150300118107846437701158064808916492244872560821

%N 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 Approaches 1.5028368...^(2^n), see A077496. Row sums of A065329 as square array. - _Henry Bottomley_, Oct 29 2001. Also row sum of square array A073345 (AK).

%D I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162.

%D I. M. H. Etherington, Some problems of non-associative combinations (I), Edinburgh Math. Notes, 32 (1940), pp. i-vi. Part II is by A. Erdelyi and I. M. H. Etherington, and is on pages vii-xiv of the same issue.

%D T. K. Moon, Enumerations of binary trees, types of trees and the number of reversible variable length codes, submitted to Discrete Applied Mathematics, 2000.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%H David Wasserman, <a href="/A001699/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 A. V. Aho and N. J. A. Sloane, <a href="http://neilsloane.com/doc/doubly.html">Some doubly exponential sequences</a>, Fib. Quart., 11 (1973), 429-437.

%H Mayfawny Bergmann, <a href="http://scholar.rose-hulman.edu/rhumj/vol15/iss2/1/"> Efficiency of Lossless Compression of a Binary Tree via its Minimal Directed Acyclic Graph Representation</a>. Rose-Hulman Undergraduate Mathematics Journal: Vol. 15 : Iss. 2, Article 1. (2014).

%H H. Bottomley, <a href="/A001699/a001699.gif">Illustration of initial terms</a>

%H I. M. H. Etherington, <a href="http://www.jstor.org/stable/3605743">Non-associate powers and a functional equation</a>, Math. Gaz. 21 (1937), 36-39; addendum 21 (1937), 153.

%H I. M. H. Etherington, <a href="/A000108/a000108_13.pdf">On non-associative combinations</a>, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162. [Annotated scanned copy]

%H 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 David Wasserman, <a href="/A001699/a001699.txt">Table of n, a(n) for n = 0..13</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinaryTree.html">Binary Tree</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%H <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>

%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>

%F a(n+1) = 2*a(n)*(a(0) + ... + a(n-1)) + a(n)^2.

%F a(n+1) = a(n)^2 + a(n) + a(n)*sqrt(4*a(n)-3), if n > 0.

%F a(n) = A003095(n+1) - A003095(n) = A003095(n)^2 - A003095(n) + 1. - _Henry Bottomley_, Apr 26 2001; offset of LHS corrected by _Anindya Bhattacharyya_, Jun 21 2013

%F a(n) = A059826(A003095(n-1)).

%F From _Peter Bala_, Feb 03 2017: (Start)

%F a(n) = Product_{k = 1..n} A213437(k).

%F a(n) + a(n-1) = A213437(n+1) - A213437(n). (End)

%p 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);

%t a[0] = 1; a[n_] := a[n] = 2*a[n-1]*Sum[a[k], {k, 0, n-2}] + a[n-1]^2; Table[a[n], {n, 0, 9}] (* _Jean-Fran├žois Alcover_, May 16 2012 *)

%o (PARI) {a(n) = if( n<=1, n>= 0, a(n-1) * (a(n-1) + a(n-2) + a(n-1) / a(n-2)))} /* Michael Somos, 2000 */

%Y Cf. A002658, A056207, A002449, A003095.

%Y Cf. A004019, A077496, A076949, A213437.

%K nonn,easy,core,nice

%O 0,3

%A _N. J. A. Sloane_, _Jeffrey Shallit_

%E Minor edits by _Vaclav Kotesovec_, Oct 04 2014

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Last modified February 18 17:08 EST 2018. Contains 299325 sequences. (Running on oeis4.)