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%I M1535 N0599 #40 Oct 08 2015 15:21:21
%S 2,5,21,61,214,669,2240,7330,24695,83257,284928,981079,3410990,
%T 11937328,42075242,149171958,531866972,1905842605,6861162880,
%U 24805692978,90035940227,327987890608,1198853954688,4395797189206,16165195705544,59609156824273,220373268471398,816677398144221
%N Number of asymmetrical dissections of n-gon.
%C This sequence, U_n in Guy's 1958 paper, counts triangulations of a regular n-gon into n-2 triangles with no nonidentity symmetries. Triangulations related by a symmetry of the underlying n-gon do not count as distinct. - _Joseph Myers_, Jun 21 2012
%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 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).
%H Joseph Myers, <a href="/A000131/b000131.txt">Table of n, a(n) for n = 7..1000</a>
%H S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="http://dx.doi.org/10.1021/ci00026a012">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
%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]
%F a(n) = (Catalan(n-2) - (n/2)*Catalan(n/2 - 1) - n*Catalan(floor(n/2) - 1) - (n/3)*Catalan(n/3 - 1) + n*Catalan(n/4 - 1) + n*Catalan(n/6 - 1))/(2*n), where Catalan(x) = 0 for noninteger x (derived from Guy's 1958 paper). - _Joseph Myers_, Jun 21 2012
%t catalan[n_] := Block[{c = Binomial[2 n, n]/(n + 1)}, If[IntegerQ[c], c, 0]]; f[n_] := (catalan[n - 2] - (n/2) catalan[n/2 - 1] - n*catalan[Floor[n/2] - 1] - (n/3)*catalan[n/3 - 1] + n*catalan[n/4 - 1] + n*catalan[n/6 - 1])/(2 n); Array[f, 28, 7] (* _Robert G. Wilson v_, Jun 23 2014 *)
%o (PARI) C(n)=if(denominator(n)==1,binomial(2*n,n)/(n+1),0)
%o a(n)=(C(n-2)/n-C(n/2-1)/2-C(n\2-1)-C(n/3-1)/3+C(n/4-1)+C(n/6-1))/2 \\ _Charles R Greathouse IV_, Apr 05 2013
%Y Cf. A000063.
%K nonn
%O 7,1
%A _N. J. A. Sloane_
%E Extended by _Joseph Myers_, Jun 21 2012
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