cp's OEIS Frontend

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).

%I A067955 #34 Sep 16 2022 07:34:02
%S A067955 1,0,0,1,0,1,3,1,8,13,15,56,79,157,399,624,1448,3061,5571,12826,25559,
%T A067955 51608,113828,227954,482591,1031681,2117323,4542331,9591243,20119244,
%U A067955 43164172,91165297,193826856,415024053,881294603,1886458874,4038398755
%N 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).
%C A067955 Number of ordered trees with n-1 edges, all of whose nodes have odd outdegree greater than two.
%C A067955 Conjecture: Number of lattice paths that do not cross below the x-axis from (1,0) to (n,0) using up-step (1,1) and down-steps {(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
%H A067955 Robert Israel, <a href="/A067955/b067955.txt">Table of n, a(n) for n = 1..2604</a>
%F A067955 a(n) = (1/n)Sum_{j=1..floor((n-1)/3)} binomial(n, j)binomial((n-3-j)/2, j-1). [formula seems wrong]
%F A067955 G.f. G(z) satisfies (1+z)*G^3 - z*G^2 - G + z = 0.
%F A067955 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
%F A067955 G.f. is the series reversion of (x-x^3)/(1-x^2+x^3). - _Joerg Arndt_, Sep 28 2015
%e A067955 a(7)= 3 because the only dissections with 7 edges are given by a hexagon dissected by any of the three halving diagonals.
%p A067955 Order := 40: solve(series((G-G^3)/(1-G^2+G^3),G)=z,G);
%p A067955 # Alternative:
%p A067955 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):
%p A067955 map(f, [$1..100]); # _Robert Israel_, Sep 01 2015
%t A067955 CoefficientList[InverseSeries[(x-x^3)/(1-x^2+x^3) + O[x]^40, x], x] // Rest (* _Jean-François Alcover_, Sep 16 2022 *)
%o A067955 (PARI) Vec(serreverse((x-x^3)/(1-x^2+x^3)+O(x^44))) \\ _Joerg Arndt_, Sep 28 2015
%K A067955 nonn
%O A067955 1,7
%A A067955 _Emeric Deutsch_, Mar 06 2002