A007160 Number of diagonal dissections of a convex (n+6)-gon into n regions. (Formerly M5094)

1, 20, 225, 1925, 14014, 91728, 556920, 3197700, 17587350, 93486536, 483367885, 2442687975, 12109051500, 59053512000, 283963030560, 1348824395160, 6338392712550, 29503515951000, 136173391604250, 623760137794794, 2837765901615876, 12830687578253600, 57687379921990000

Offset

1,2

Comments

Number of standard tableaux of shape (n,n,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+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.

a(n) = number of noncrossing partitions of 2n+4 into n blocks all of size at least 2. (End)

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

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

Petr Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation, Vol. 20, No. 5-6 (1995), 595-601.

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.

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

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

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.

From Amiram Eldar, Nov 04 2025: (Start)

a(n) ~ 2^(2*n+1) * n^(5/2) / (3 * sqrt(Pi)).

Sum_{n>=1} 1/a(n) = 1795/3 - 88*sqrt(3)*Pi - 12*Pi^2.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1349/3 - 448*sqrt(5)*log(phi) + 144*log(phi)^2, where phi is the golden ratio (A001622). (End)

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 *)

Prog

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

Crossrefs

A diagonal of A033282.

Cf. A001622.

Sequence in context: A178261 A054329 A112503 * A195265 A321674 A264876

Adjacent sequences: A007157 A007158 A007159 * A007161 A007162 A007163

Keyword

easy,nonn,nice

Author

N. J. A. Sloane

Extensions

Offset is correct!

Status

approved