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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)
1, 1, 3, 21, 651, 457653, 210065930571, 44127887745696109598901, 1947270476915296449559659317606103024276803403, 3791862310265926082868235028027893277370233150300118107846437701158064808916492244872560821 (list; graph; refs; listen; history; text; internal format)



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).


Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 307.

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

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.

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

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

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


David Wasserman, Table of n, a(n) for n = 0..12 [Shortened file because terms grow rapidly: see Wasserman link below for an additional term]

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart., 11 (1973), 429-437.

Mayfawny Bergmann, Efficiency of Lossless Compression of a Binary Tree via its Minimal Directed Acyclic Graph Representation. Rose-Hulman Undergraduate Mathematics Journal: Vol. 15 : Iss. 2, Article 1. (2014).

H. Bottomley, Illustration of initial terms

I. M. H. Etherington, Non-associate powers and a functional equation, Math. Gaz. 21 (1937), 36-39; addendum 21 (1937), 153.

I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162. [Annotated scanned copy]

C. Lenormand, Arbres et permutations II, see p. 6

David Wasserman, Table of n, a(n) for n = 0..13

Eric Weisstein's World of Mathematics, Binary Tree

Index entries for "core" sequences

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

Index entries for sequences related to parenthesizing

Index entries for sequences of form a(n+1)=a(n)^2 + ...


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

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

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

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

From Peter Bala, Feb 03 2017: (Start)

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

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


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);


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 *)


(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 */


Cf. A002658, A056207, A002449, A003095.

Cf. A004019, A077496, A076949, A213437.

Sequence in context: A093549 A012044 A098918 * A291967 A171104 A057600

Adjacent sequences:  A001696 A001697 A001698 * A001700 A001701 A001702




N. J. A. Sloane, Jeffrey Shallit


Minor edits by Vaclav Kotesovec, Oct 04 2014



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Last modified January 19 11:47 EST 2019. Contains 319306 sequences. (Running on oeis4.)