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

0,2

Comments

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

From Oliver Pechenik, May 02 2014: (Start)

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.

Number of noncrossing partitions of 2n+7 into n+1 blocks all of size at least 2. (End)

Links

Table of n, a(n) for n=0..21.

David Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, Vol. 105, No. 3 (1998), 256-257.

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

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

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

Hua Xin, Lattice points of flow polytopes related to caracol graphs, AIMS Elect. Res. Archive 33(10) (2025) 6141-6175. See p. 6159, Table 4.

Formula

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

From R. J. Mathar, Feb 09 2020: (Start)

G.f.: 3F2(4,6,9/2 ; 2,8 ; 4*x).

D-finite with recurrence n*(n+7)*(n+1)*a(n) - 2*(n+5)*(n+3)*(2*n+7)*a(n-1) = 0. (End)

From Amiram Eldar, Nov 04 2025: (Start)

a(n) ~ 2^(2*n+4) * n^(7/2) / (15 * sqrt(Pi)).

Sum_{n>=0} 1/a(n) = 27574/9 - 450*sqrt(3)*Pi - 560*Pi^2/9.

Sum_{n>=0} (-1)^n/a(n) = 11866/9 - 3260*sqrt(5)*log(phi)/3 - 640*log(phi)^2, where phi is the golden ratio (A001622). (End)

Mathematica

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

Prog

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

Crossrefs

Cf. A001622.

Sequence in context: A321954 A000535 A251770 * A125462 A326605 A296853

Adjacent sequences: A033277 A033278 A033279 * A033281 A033282 A033283

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Michel Marcus, Jun 18 2015

Status

approved