A002058 Number of internal triangles in all triangulations of an (n+1)-gon.

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

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn

Comments

From Richard Stanley, Jan 30 2014: (Start)

The previous name "Number of partitions of a n-gon into (n-3) 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

• A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262
• A. Cayley, On the partitions of a polygon, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.

Crossrefs

Cf. A002059, A002060.

Programs

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

Formula

a(n) = 2*binomial(2*n-5,n-5) = 2*A003516(n-3). - David Callan, Mar 30 2007

G.f. 64*x^5/((1+sqrt(1-4*x))^5*sqrt(1-4*x)). - R. J. Mathar, Nov 27 2011

a(n) ~ 4^n/(16*sqrt(Pi*n)). - Ilya Gutkovskiy, Apr 11 2017

Extensions

Definition corrected by Richard Stanley, Jan 30 2014