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Search: id:A000150
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| A000150 |
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Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge.
(Formerly M1753 N0696)
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6
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| 0, 0, 1, 2, 7, 20, 66, 212, 715, 2424, 8398, 29372, 104006, 371384, 1337220, 4847208, 17678835, 64821680, 238819350, 883629164, 3282060210, 12233125112, 45741281820, 171529777432, 644952073662, 2430973096720, 9183676536076
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Number of Dyck paths of length 2n having an odd number of peaks at even height. Example: a(3)=2 because we have UDU(UD)D and U(UD)DUD, where U=(1,1), D=(1,-1) and the peaks at even height are shown between parentheses. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 13 2004
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REFERENCES
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S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751
R. K. Guy, ``Dissecting a polygon into triangles,'' Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.
F. Harary and E. M. Palmer, On acyclic simplicial complexes. Mathematika 15 1968 115-122.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.26).
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).
P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
Index entries for sequences related to Lyndon words
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FORMULA
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Let c(x) = (1-sqrt(1-4*x))/(2*x) = g.f. for Catalans (A000108), let d(x) = 1+x*c(x^2). Then g.f. is (c(x)-d(x))/2.
G.f.=[sqrt(1-4z^2)-sqrt(1-4z)-2z]/(4z). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 13 2004
a(n) = ( 2^(n-3)/sqrt(Pi) ) * ( 4*2^n*GAMMA(n+1/2)/GAMMA(n+2) + ((-1)^n - 1)*GAMMA(n/2)/GAMMA(n/2 + 3/2) ) for n>0 [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 11 2009]
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CROSSREFS
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a(n) = T(2n+2, n), array T as in A051168, a count of Lyndon words.
Cf. A051168, A005430.
Cf. A007595.
A diagonal of the square array described in A051168.
Sequence in context: A035071 A055891 A122877 this_sequence A115117 A029890 A095268
Adjacent sequences: A000147 A000148 A000149 this_sequence A000151 A000152 A000153
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KEYWORD
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nonn,nice,easy,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Additional comments from Clark Kimberling (ck6(AT)evansville.edu)
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