A033276 Number of diagonal dissections of an n-gon into 4 regions.

0, 14, 84, 300, 825, 1925, 4004, 7644, 13650, 23100, 37400, 58344, 88179, 129675, 186200, 261800, 361284, 490314, 655500, 864500, 1126125, 1450449, 1848924, 2334500, 2921750, 3627000, 4468464, 5466384, 6643175, 8023575, 9634800, 11506704, 13671944

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of standard tableaux of shape (n-4,2,2,2) (n>=6). - Emeric Deutsch, May 20 2004

Number of short bushes with n+2 edges and 4 branch nodes (i.e. nodes with outdegree at least 2). A short bush is an ordered tree with no nodes of outdegree 1. Example: a(6)=14 because the only short bushes with 8 edges and 4 branch nodes are the fourteen full binary trees with 8 edges. Column 4 of A108263. - Emeric Deutsch, May 29 2005

Links

• Vincenzo Librandi, Table of n, a(n) for n = 5..1000
• David Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, Vol. 105, No. 3 (1998), 256-257.
• Frank R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., Vol. 204, No. 1-3 (1999), 73-112.
• Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978), 370-388, Table 1.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A033275, A108263.

Programs

Magma

[(Binomial(n+2,3)*Binomial(n-3,3))/4: n in [5..50]]; // Vincenzo Librandi, Mar 15 2014

Mathematica

Table[(Binomial[n+2,3]Binomial[n-3,3])/4,{n,5,40}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,14,84,300,825,1925,4004},40] (* Harvey P. Dale, Mar 13 2014 *)
CoefficientList[Series[x (14 - 14 x + 6 x^2 - x^3)/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *)

Formula

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

G.f.: x^6*(14-14x+6x^2-x^3)/(1-x)^7. - Emeric Deutsch, May 29 2005

From Amiram Eldar, Aug 30 2022: (Start)

Sum_{n>=6} 1/a(n) = 109/1225.

Sum_{n>=6} (-1)^n/a(n) = 192*log(2)/35 - 4582/1225. (End)

Extensions

More terms from Vincenzo Librandi, Mar 15 2014