A033275 Number of diagonal dissections of an n-gon into 3 regions.

0, 5, 21, 56, 120, 225, 385, 616, 936, 1365, 1925, 2640, 3536, 4641, 5985, 7600, 9520, 11781, 14421, 17480, 21000, 25025, 29601, 34776, 40600, 47125, 54405, 62496, 71456, 81345, 92225, 104160, 117216, 131461, 146965, 163800, 182040, 201761, 223041, 245960

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of standard tableaux of shape (n-3,2,2) (n>=5). - Emeric Deutsch, May 13 2004

Number of short bushes with n+1 edges and 3 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(5)=5 because the only short bushes with 6 edges and 3 branch nodes are the five full binary trees with 6 edges. Column 3 of A108263. - Emeric Deutsch, May 29 2005

Links

• Indranil Ghosh, Table of n, a(n) for n = 4..10000
• David Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, Vol. 105, No. 3 (1998), pp. 256-257.
• Frank R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., Vol. 204, No. 1-3 (1999), pp. 73-112.
• Ronald C. Read, On general dissections of a polygon, Aequat. Math., Vol. 18 (1978), pp. 370-388, Table 1.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

2nd skew subdiagonal of A033282.

Cf. A033276, A108263.

Programs

Mathematica

a[4]=0; a[n_]:=Binomial[n+1,2]*Binomial[n-3,2]/3; Table[a[n],{n,4,43}] (* Indranil Ghosh, Feb 20 2017 *)

PARI

concat(0, Vec(z^5*(5-4*z+z^2)/(1-z)^5 + O(z^60))) \\ Michel Marcus, Jun 18 2015

PARI

a(n) = binomial(n+1, 2)*binomial(n-3, 2)/3 \\ Charles R Greathouse IV, Feb 20 2017

Sage

def A033275(n): return (binomial(n+1, 2)*binomial(n-3, 2))//3
print([A033275(n) for n in range(4,50)]) # Peter Luschny, Apr 03 2020

Formula

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

G.f.: z^5*(5-4*z+z^2)/(1-z)^5. - Emeric Deutsch, May 29 2005

From Amiram Eldar, Jan 06 2021: (Start)

Sum_{n>=5} 1/a(n) = 43/150.

Sum_{n>=5} (-1)^(n+1)/a(n) = 16*log(2)/5 - 154/75. (End)

E.g.f.: x*(exp(x)*(12 - 6*x + x^3) - 6*(2 + x))/12. - Stefano Spezia, Feb 21 2024