%I A000150 M1753 N0696
%S A000150 0,0,1,2,7,20,66,212,715,2424,8398,29372,104006,371384,1337220,
%T A000150 4847208,17678835,64821680,238819350,883629164,3282060210,12233125112,
%U A000150 45741281820,171529777432,644952073662,2430973096720,9183676536076
%N A000150 Number of dissections of an n-gon, rooted at an exterior edge, asymmetric 
               with respect to that edge.
%C A000150 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
%D A000150 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
%D A000150 R. K. Guy, ``Dissecting a polygon into triangles,'' Bull. Malayan Math. 
               Soc., Vol. 5, pp. 57-60, 1958.
%D A000150 F. Harary and E. M. Palmer, On acyclic simplicial complexes. Mathematika 
               15 1968 115-122.
%D A000150 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, p. 78, (3.5.26).
%D A000150 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000150 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000150 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.
%H A000150 T. D. Noe, <a href="b000150.txt">Table of n, a(n) for n=0..200</a>
%H A000150 <a href="Sindx_Lu.html#Lyndon">Index entries for sequences related to 
               Lyndon words</a>
%F A000150 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.
%F A000150 G.f.=[sqrt(1-4z^2)-sqrt(1-4z)-2z]/(4z). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Nov 13 2004
%F A000150 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]
%Y A000150 a(n) = T(2n+2, n), array T as in A051168, a count of Lyndon words.
%Y A000150 Cf. A051168, A005430.
%Y A000150 Cf. A007595.
%Y A000150 A diagonal of the square array described in A051168.
%Y A000150 Sequence in context: A035071 A055891 A122877 this_sequence A115117 A029890 
               A095268
%Y A000150 Adjacent sequences: A000147 A000148 A000149 this_sequence A000151 A000152 
               A000153
%K A000150 nonn,nice,easy,new
%O A000150 0,4
%A A000150 N. J. A. Sloane (njas(AT)research.att.com).
%E A000150 Additional comments from Clark Kimberling (ck6(AT)evansville.edu)