A007160 Number of diagonal dissections of a convex (n+6)-gon into n regions.

1, 20, 225, 1925, 14014, 91728, 556920, 3197700, 17587350, 93486536, 483367885, 2442687975, 12109051500, 59053512000, 283963030560, 1348824395160, 6338392712550, 29503515951000, 136173391604250

Offset: 1

Author

N. J. A. Sloane

Keywords

• easy
• nonn
• nice

Comments

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

Number of increasing tableaux of shape (n+4,n+4) with largest entry 2n+4. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, such that the set of entries forms an initial segment of the positive integers. - Oliver Pechenik, May 02 2014

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

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..150
• D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
• 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].
• O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory A, 125 (2014), 357-378.
• 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.

Crossrefs

A diagonal of A033282.

Programs

Magma

[Binomial(n+3, 4)*Binomial(2*n+4, n-1)/n  : n in [1..30]]; // Vincenzo Librandi, Nov 17 2011

Mathematica

a[n_] := (n+1)(n+2)(n+3)*Binomial[2n+4, n-1]/24; Table[a[n], {n, 1, 19}](* Jean-François Alcover, Nov 16 2011 *)

Formula

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

a(n) = binomial(n+3, 4)*binomial(2n+4, n-1)/n.

Extensions

Offset is correct!