A033280 Number of diagonal dissections of a convex (n+8)-gon into n+1 regions.

1, 27, 385, 4004, 34398, 259896, 1790712, 11511720, 70114902, 409003595, 2303105805, 12593413560, 67173369900, 350777861280, 1798432526880, 9073909567440, 45140379405030, 221768094898350, 1077403874372826, 5182007298602904, 24699073588138180, 116759256962107760

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Number of increasing tableaux of shape (n+6,n+6) 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+1 blocks all of size at least 2. - Oliver Pechenik, May 02 2014

Links

• D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
• 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. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

Programs

Mathematica

Table[(Binomial[n+5,5]Binomial[2n+7,n])/(n+1),{n,0,30}] (* Harvey P. Dale, Oct 16 2016 *)

PARI

vector(30, n, n--; binomial(n+5, 5)*binomial(2*n+7, n)/(n+1)) \\ Michel Marcus, Jun 18 2015

Formula

a(n) = binomial(n+5, 5)*binomial(2n+7, n)/(n+1).

G.f.: 3F2(4,6,9/2 ; 2,8 ; 4*x). - R. J. Mathar, Feb 09 2020

D-finite with recurrence n*(n+7)*(n+1)*a(n) -2*(n+5)*(n+3)*(2*n+7)*a(n-1)=0. - R. J. Mathar, Feb 09 2020

Extensions

More terms from Michel Marcus, Jun 18 2015