A002058 Number of internal triangles in all triangulations of an (n+1)-gon. (Formerly M2069 N0817)

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

Offset

5,1

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

Table of n, a(n) for n=5..27.

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.

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

Prog

(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

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