

A002058


Number of internal triangles in all triangulations of an (n+1)gon.
(Formerly M2069 N0817)


6



2, 14, 72, 330, 1430, 6006, 24752, 100776, 406980, 1634380, 6537520, 26075790, 103791870, 412506150, 1637618400, 6495886320, 25751549340, 102042235620, 404225281200, 1600944863700, 6339741660252, 25103519174844, 99399793096352
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OFFSET

5,1


COMMENTS

From Richard Stanley, Jan 30 2014: (Start)
The previous name "Number of partitions of a ngon into (n3) parts" was erroneous.
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.
(End)
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.


REFERENCES

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).


LINKS

Table of n, a(n) for n=5..27.
A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237262
A. Cayley, On the partitions of a polygon, Collected Mathematical Papers. Vols. 113, Cambridge Univ. Press, London, 18891897, Vol. 13, pp. 93ff.


FORMULA

a(n) = 2*binomial(2*n5,n5) = 2*A003516(n3).  David Callan, Mar 30 2007
G.f. 64*x^5/((1+sqrt(14*x))^5*sqrt(14*x)).  R. J. Mathar, Nov 27 2011
a(n) ~ 4^n/(16*sqrt(Pi*n)).  Ilya Gutkovskiy, Apr 11 2017


PROG

(PARI) x='x+O('x^66); Vec(64*x^5/((1+sqrt(14*x))^5*sqrt(14*x))) \\ Joerg Arndt, Jan 30 2014


CROSSREFS

Cf. A002059, A002060.
Sequence in context: A072888 A171012 A094583 * A095933 A263218 A189305
Adjacent sequences: A002055 A002056 A002057 * A002059 A002060 A002061


KEYWORD

nonn


AUTHOR

N. J. A. Sloane.


EXTENSIONS

Definition corrected by Richard Stanley, Jan 30 2014


STATUS

approved



