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 A000150 Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge. (Formerly M1753 N0696) 8

%I M1753 N0696

%S 0,0,1,2,7,20,66,212,715,2424,8398,29372,104006,371384,1337220,

%T 4847208,17678835,64821680,238819350,883629164,3282060210,12233125112,

%U 45741281820,171529777432,644952073662,2430973096720,9183676536076

%N Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge.

%C 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_, Nov 13 2004

%C For n>=1, a(n) is the number of unordered binary trees with n internal nodes in which the left subtree is distinct from the right subtree. - _Geoffrey Critzer_, Feb 21 2013

%C Assuming offset -1 this is an analog of A275166: pairs of distinct Catalan numbers with index sum n. - _R. J. Mathar_, Jul 19 2016

%D 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 R. K. Guy, "Dissecting a polygon into triangles," Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.

%D R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967.

%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.26).

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

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

%D 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 T. D. Noe, <a href="/A000150/b000150.txt">Table of n, a(n) for n=0..200</a>

%H S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="/A002057/a002057.pdf">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

%H R. K. Guy, <a href="/A000108/a000108_11.pdf">Dissecting a polygon into triangles</a>, Research Paper #9, Math. Dept., Univ. Calgary, 1967. [Annotated scanned copy]

%H F. Harary and E. M. Palmer, <a href="http://dx.doi.org/10.1112/S002557930000245X">On acyclic simplicial complexes</a>, Mathematika 15 1968 115-122.

%H <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>

%F Let c(x) = (1-sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers (A000108), let d(x) = 1+x*c(x^2). Then g.f. is (c(x)-d(x))/2.

%F G.f.: (sqrt(1-4*z^2) - sqrt(1-4*z) - 2*z)/(4*z). - _Emeric Deutsch_, Nov 13 2004

%F With c(x) defined as above: g.f. = x*(c(x)^2/2 - c(x^2)/2). - _Geoffrey Critzer_, Feb 21 2013

%F 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. - _Mark van Hoeij_, Nov 11 2009

%F a(n) ~ 2^(2*n-1) / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 10 2014

%F a(2n) = A000108(2n) / 2; a(2n+1) = ( A000108(2n+1) - A000108(n) ) / 2. - _John Bodeen_, Jun 24 2015

%t nn=20;CoefficientList[Series[x/2(((1-(1-4x)^(1/2))/(2x))^2-(1-(1-4x^2)^(1/2))/(2x^2)),{x,0,nn}],x] (* _Geoffrey Critzer_, Feb 21 2013 *)

%Y a(n) = T(2n+2, n), array T as in A051168, a count of Lyndon words.

%Y Cf. A051168, A005430.

%Y Cf. A007595.

%Y A diagonal of the square array described in A051168.

%K nonn,nice,easy

%O 0,4

%A _N. J. A. Sloane_