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
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).
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
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 *)
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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved