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%I M2069 N0817 #33 Dec 19 2021 10:05:25
%S 2,14,72,330,1430,6006,24752,100776,406980,1634380,6537520,26075790,
%T 103791870,412506150,1637618400,6495886320,25751549340,102042235620,
%U 404225281200,1600944863700,6339741660252,25103519174844,99399793096352
%N Number of internal triangles in all triangulations of an (n+1)-gon.
%C From _Richard Stanley_, Jan 30 2014: (Start)
%C The previous name "Number of partitions of a n-gon into (n-3) parts" was erroneous.
%C Cayley does not seem to have a combinatorial interpretation of this sequence. He just uses it as an auxiliary sequence, nor am I aware of a combinatorial interpretation in the literature.
%C (End)
%C First subdiagonal of the table of V(r,k) on page 240. The values V(11,8) = 24052, V(13,10)= 396800 and V(15,12)= 6547520 of the publication are replaced/corrected in the sequence.
%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 A. Cayley, <a href="http://dx.doi.org/10.1112/plms/s1-22.1.237">On the partitions of a polygon</a>, Proc. London Math. Soc., 22 (1891), 237-262
%H A. Cayley, <a href="http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?sid=b88432273f115fb346725f1a42422e19&idno=abs3153.0013.001&c=umhistmath&cc=umhistmath&seq=110&view=image">On the partitions of a polygon</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
%F a(n) = 2*binomial(2*n-5,n-5) = 2*A003516(n-3). - _David Callan_, Mar 30 2007
%F G.f. 64*x^5/((1+sqrt(1-4*x))^5*sqrt(1-4*x)). - _R. J. Mathar_, Nov 27 2011
%F a(n) ~ 4^n/(16*sqrt(Pi*n)). - _Ilya Gutkovskiy_, Apr 11 2017
%o (PARI) x='x+O('x^66); Vec(64*x^5/((1+sqrt(1-4*x))^5*sqrt(1-4*x))) \\ _Joerg Arndt_, Jan 30 2014
%Y Cf. A002059, A002060.
%K nonn
%O 5,1
%A _N. J. A. Sloane_
%E Definition corrected by _Richard Stanley_, Jan 30 2014
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