A002056 Number of diagonal dissections of a convex n-gon into n-5 regions. (Formerly M4941 N2115)

1, 14, 120, 825, 5005, 28028, 148512, 755820, 3730650, 17978180, 84987760, 395482815, 1816357725, 8250123000, 37119350400, 165645101160, 733919156190, 3231337461300, 14147884842000, 61636377252450, 267325773340626, 1154761882042824, 4969989654817600

Offset

6,2

Comments

Number of standard tableaux of shape (n-5,n-5,1,1,1) (see Stanley reference). - Emeric Deutsch, May 20 2004

Number of increasing tableaux of shape (n-2,n-2) with largest entry 2n-7. 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

Number of noncrossing partitions of 2n-7 into n-5 blocks all 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=6..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.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics; arXiv:0912.0072 [math.NT], 2009.

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

Ronald C. 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, 3)*binomial(2n-7, n-6)/(n-5).

G.f.: (x-1+(1-11*x+40*x^2-50*x^3+10*x^4)*(1-4*x)^(-5/2))/(2*x^5). - Mark van Hoeij, Oct 25 2011

a(n) ~ 4^n*n^(3/2)/(768*sqrt(Pi)). - Ilya Gutkovskiy, Apr 11 2017

D-finite with recurrence: -(n-1)*(n-5)*(n-6)*a(n) +2*(2*n-7)*(n-3)*(n-4)*a(n-1)=0. - R. J. Mathar, Feb 16 2020

Maple

A002056:=n->binomial(n-3, 3)*binomial(2*n-7, n-6)/(n-5): seq(A002056(n), n=6..40); # Wesley Ivan Hurt, Apr 12 2017

Mathematica

Table[Binomial[n - 3, 3] Binomial[2n - 7, n - 6]/(n - 5), {n, 6, 50}] (* Indranil Ghosh, Apr 11 2017 *)

Prog

(Magma) [Binomial(n-3, 3)*Binomial(2*n-7, n-6)/(n-5): n in [6..30]]; // Vincenzo Librandi, Feb 18 2020

Crossrefs

Sequence in context: A202072 A280003 A004312 * A249980 A206635 A163942

Adjacent sequences: A002053 A002054 A002055 * A002057 A002058 A002059

Keyword

nonn

Author

N. J. A. Sloane

Status

approved