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A002055 Number of diagonal dissections of a convex n-gon into n-4 regions.
(Formerly M4639 N1982)
9
1, 9, 56, 300, 1485, 7007, 32032, 143208, 629850, 2735810, 11767536, 50220040, 212952285, 898198875, 3771484800, 15775723920, 65770848990, 273420862110, 1133802618000, 4691140763400, 19371432850770, 79850555673174 (list; graph; refs; listen; history; text; internal format)
OFFSET

5,2

COMMENTS

Number of standard tableaux of shape (n-4,n-4,1,1) (see Stanley reference). - Emeric Deutsch, May 20 2004

Number of increasing tableaux of shape (n-2,n-2) with largest entry 2n-6. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. - Oliver Pechenik, May 02 2014

a(n) = number of noncrossing partitions of 2n-6 into n-4 blocks, each of size at least 2. - Oliver Pechenik, May 02 2014

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

T. D. Noe, Table of n, a(n) for n = 5..100

D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.

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

P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.

O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, arXiv:1209.1355 [math.CO], 2012-2014.

O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory A, 125 (2014), 357-378.

R. C. Read, On general dissections of a polygon, Preprint (1974)

C. R. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388, Table 1.

R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

FORMULA

a(n) = binomial(n-3, 2)*binomial(2*n-6, n-5)/(n-4).

With offset 0, this has a(n)=(n+2)*C(2n+4,n)/2 and e.g.f. dif(dif(x*dif(exp(2x)*Bessel_I(2,2x),x),x),x)/2. - Paul Barry, Aug 25 2007

G.f.: 16*x^5*(x+sqrt(1-4x))/((1-4x)^(3/2) *(1+sqrt(1-4x))^4 ). - R. J. Mathar, Nov 17 2011

Conjecture: (n-1)*a(n) +(23-11n)*a(n-1) +10*(4n-13)*a(n-2) +10*(23-5n)*a(n-3) +4*(2n-13)*a(n-4)=0. - R. J. Mathar, Nov 17 2011

a(n) ~ 4^n*sqrt(n)/(128*sqrt(Pi)). - Ilya Gutkovskiy, Apr 11 2017

MATHEMATICA

Table[(Binomial[n-3, 2]Binomial[2n-6, n-5])/(n-4), {n, 5, 30}] (* Harvey P. Dale, Nov 06 2011 *)

PROG

(PARI) a(n) = (binomial(n - 3, 2) * binomial(2*n - 6, n - 5))/(n - 4);

for(n=5, 30, print1(a(n), ", ")) \\ Indranil Ghosh, Apr 11 2017

CROSSREFS

a(n) = f(n,n+1) where f is given in A034261.

Sequence in context: A196861 A211844 A172065 * A026842 A026846 A026849

Adjacent sequences:  A002052 A002053 A002054 * A002056 A002057 A002058

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 13 19:25 EST 2018. Contains 317149 sequences. (Running on oeis4.)