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A033275 Number of diagonal dissections of an n-gon into 3 regions. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,2

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

D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.

R. 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 (5,-10,10,-5,1).

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

MATHEMATICA

f[n_]:=n*(n+2)*(n+4)/3; s=0; lst={}; Do[AppendTo[lst, s+=f[n]], {n, 0, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 08 2009 *)

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 *)

PROG

(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

(Python)

import math

def C(n, r):

....f=math.factorial

....return f(n)/f(r)/f(n-r)

def A033275(n):

....if n==4:return '0'

....return str(C(n+1, 2)*C(n-3, 2)/3) # Indranil Ghosh, Feb 20 2017

CROSSREFS

2nd skew subdiagonal of A033282.

Cf. A033276, A108263.

Sequence in context: A122244 A146854 A299120 * A166464 A059859 A146617

Adjacent sequences:  A033272 A033273 A033274 * A033276 A033277 A033278

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 22 08:06 EDT 2019. Contains 322329 sequences. (Running on oeis4.)