

A067955


Number of dissections of a convex polygon by nonintersecting diagonals into polygons with even number of sides and having a total number of n edges (sides and diagonals).


3



1, 0, 0, 1, 0, 1, 3, 1, 8, 13, 15, 56, 79, 157, 399, 624, 1448, 3061, 5571, 12826, 25559, 51608, 113828, 227954, 482591, 1031681, 2117323, 4542331, 9591243, 20119244, 43164172, 91165297, 193826856, 415024053, 881294603, 1886458874, 4038398755
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OFFSET

1,7


COMMENTS

Number of ordered trees with n1 edges, all of whose nodes have odd outdegree greater than two.
Conjecture: Number of lattice paths that do not cross below the xaxis from (1,0) to (n,0) using upstep (1,1) and downsteps {(1,z): z is a positive even integer}. For example, a(8) = 1: [(1,1)(1,1)(1,1)(1,1)(1,1)(1,1)(1,6)].  Nicholas Ham, Aug 24 2015


LINKS

Robert Israel, Table of n, a(n) for n = 1..2604


FORMULA

a(n) = (1/n)Sum_{j=1..floor((n1)/3)} binomial(n, j)binomial((n3j)/2, j1). [formula seems wrong]
G.f. G(z) satisfies (1+z)*G^3  z*G^2  G + z = 0.
115*n*(n+1)*a(n)+(617*n+1236)*(n+1)*a(n+1)+(2*(569*n^2+2657*n+3006))*a(n+2)+(2*(436*n^2+2737*n+4254))*a(n+3)+(6*(32*n^2+267*n+554))*a(n+4)(4*(29*n^2+260*n+570))*a(n+5)(8*(n+6))*(11*n+53)*a(n+6)(16*(n+7))*(n+6)*a(n+7) = 0.  Robert Israel, Sep 01 2015
G.f. is the series reversion of (xx^3)/(1x^2+x^3).  Joerg Arndt, Sep 28 2015


EXAMPLE

a(7)= 3 because the only dissections with 7 edges are given by a hexagon dissected by any of the three halving diagonals.


MAPLE

Order := 40: solve(series((GG^3)/(1G^2+G^3), G)=z, G);
# Alternative:
f:= gfun:rectoproc({115*n*(n+1)*a(n)+(617*n+1236)*(n+1)*a(n+1)+(2*(569*n^2+2657*n+3006))*a(n+2)+(2*(436*n^2+2737*n+4254))*a(n+3)+(6*(32*n^2+267*n+554))*a(n+4)(4*(29*n^2+260*n+570))*a(n+5)(8*(n+6))*(11*n+53)*a(n+6)(16*(n+7))*(n+6)*a(n+7) = 0, a(0)=0, a(1)=1, a(2)=0, a(3)=0, a(4)=1, a(5)=0, a(6)=1}, a(n), remember):
map(f, [$1..100]); # Robert Israel, Sep 01 2015


PROG

(PARI) Vec(serreverse((xx^3)/(1x^2+x^3)+O(x^44))) \\ Joerg Arndt, Sep 28 2015


CROSSREFS

Sequence in context: A102537 A131202 A287987 * A182509 A049965 A221736
Adjacent sequences: A067952 A067953 A067954 * A067956 A067957 A067958


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Mar 06 2002


STATUS

approved



