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A002449
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Number of different types of binary trees of height n.
(Formerly M1683 N0664)
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14
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1, 1, 2, 6, 26, 166, 1626, 25510, 664666, 29559718, 2290267226, 314039061414, 77160820913242, 34317392762489766, 27859502236825957466, 41575811106337540656038, 114746581654195790543205466
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Equals the number of partitions of 2^n-1 into powers of 2 (cf. A018819). a(n) = A018819(2^n-1) = binary partitions of 2^n-1. - Paul D. Hanna (pauldhanna(AT)juno.com), Sep 22 2004
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REFERENCES
| George E. Andrews, Peter Paule, Axel Riese and Volker Strehl, "MacMahon's Partition Analysis V: Bijections, recursions and magic squares," in Algebraic Combinatorics and Applications, edited by Anton Betten, Axel Kohnert, Reinhard Laue and Alfred Wassermann [Proceedings of ALCOMA, September 1999] (Springer, 2001), 1-39.
A. Cayley, "On a problem in the partition of numbers," Philosophical Magazine (4) 13 (1857), 245-248; reprinted in his Collected Math. Papers, Vol. 3, pp. 247-249
R. F. Churchhouse, Congruence properties of the binary partition function. Proc. Cambridge Philos. Soc. 66 1969 371-376.
R. F. Churchhouse, Binary partitions, pp. 397-400 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
D. E. Knuth, Selected Papers on Analysis of Algorithms, p. 75 (gives asymptotic formula and lower bound).
H. Minc, The free commutative entropic logarithmetic. Proc. Roy. Soc. Edinburgh Sect. A 65 1959 177-192 (1959).
T. K. Moon (tmoon(AT)artemis.ece.usu.edu), Enumerations of binary trees, types of trees and the number of reversiblevariable length codes, submitted to Discrete Applied Mathematics, 2000.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..50
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.
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FORMULA
| a(n) = A098539(n, 1). - Paul D. Hanna (pauldhanna(AT)juno.com), Sep 13 2004
G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1) + xF(x,2n) for n>0 with F(x,0)=1. - Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2007
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MAPLE
| d := proc(n) option remember; if n=0 then 1 else sum(d(n-1), k=1..2*k) fi end; A002449 := n -> eval(d(n-1), k=1);
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MATHEMATICA
| lim = 16; p[0] = p[1] = 1; Do[If[OddQ[n], p[n] = p[n - 1], p[n] = p[n - 1] + p[n/2]], {n, 1, 2^lim - 1}]; a[n_] := p[2^n - 1]; Table[a[n], {n, 0, lim}] (* From Jean-François Alcover, Sep 20 2011, after P. D. Hanna *)
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PROG
| (PARI) {a(n)=local(A, B, C, m); A=matrix(1, 1); A[1, 1]=1; for(m=2, n+1, B=A^2; C=matrix(m, m); for(j=1, m, for(k=1, j, if(j<3|k==j|k>m-1, C[j, k]=1, if(k==1, C[j, k]=B[j-1, 1], C[j, k]=B[j-1, k-1])); )); A=C); A[n+1, 1]} (Paul Hanna)
(PARI) a(n)=polcoeff(1/prod(k=0, n, 1-x^(2^k)+O(x^(2^n))), 2^n-1)
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CROSSREFS
| Cf. A001699, A056207.
Cf. A098539.
Cf. A018819.
Sequence in context: A141713 A005272 A178089 * A059430 A086584 A032014
Adjacent sequences: A002446 A002447 A002448 * A002450 A002451 A002452
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Recurrence, Maple program and more terms from Michael Kleber (michael.kleber(AT)gmail.com), Dec 05 2000
Cayley reference from D. E. Knuth, Aug 17, 2001
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