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Search: id:A001699
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| A001699 |
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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)
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+0
15
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| 1, 1, 3, 21, 651, 457653, 210065930571, 44127887745696109598901, 1947270476915296449559659317606103024276803403, 37918623102659260828682350280278932773702331503001181078464377011580648089164922\ 44872560821
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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).
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REFERENCES
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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).
I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162.
T. K. Moon, Enumerations of binary trees, types of trees and the number of reversiblevariable length codes, submitted to Discrete Applied Mathematics, 2000.
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LINKS
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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.
H. Bottomley, Illustration of initial terms
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, Link to a section of The World of Mathematics.
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 + ...
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FORMULA
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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+1) = A003095(n+1)-A003095(n) = A003095(n)^2- A003095(n)+1. - Henry Bottomley (se16(AT)btinternet.com), Apr 26 2001
a(n)=A059826(A003095(n-1))
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MAPLE
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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);
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PROGRAM
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(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
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CROSSREFS
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Cf. A002658, A056207, A002449, A003095.
Cf. A004019.
Adjacent sequences: A001696 A001697 A001698 this_sequence A001700 A001701 A001702
Sequence in context: A093549 A012044 A098918 this_sequence A162924 A057600 A079269
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KEYWORD
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nonn,easy,core,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Jeffrey Shallit
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