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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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7
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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
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OFFSET
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0,4
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COMMENTS
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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, Nov 13 2004
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
Assuming offset -1 this is an analog of A275166: pairs of distinct Catalan numbers with index sum n. - R. J. Mathar, Jul 19 2016
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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.
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967.
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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P. J. 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. [Scanned annotated and corrected copy]
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FORMULA
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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.
G.f.: (sqrt(1-4*z^2) - sqrt(1-4*z) - 2*z)/(4*z). - Emeric Deutsch, Nov 13 2004
With c(x) defined as above: g.f. = x*(c(x)^2/2 - c(x^2)/2). - Geoffrey Critzer, Feb 21 2013
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
D-finite with recurrence +n*(n+1)*(n-2)^2*a(n) -2*n*(2*n-5)*(n-1)^2*a(n-1) -4*n*(n-2)^3*a(n-2) +8*(2*n-5)*(n-3)*(n-1)^2*a(n-3)=0. - R. J. Mathar, Oct 28 2021
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MATHEMATICA
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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 *)
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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.
A diagonal of the square array described in A051168.
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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