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%I #30 Jul 19 2017 20:17:48
%S 1,0,0,0,1,1,1,1,6,12,19,27,71,166,329,579,1222,2756,5921,11754,24179,
%T 52372,114031,239726,502269,1074961,2333143,5017552,10714567,23006558,
%U 49861081,108122488,233691980,505329915,1097463037,2389325284,5199960642,11314793335,24663217250,53864633059
%N Expansion of series reversion of x*(1-x^3*sum(k>=1, x^k)).
%C Dissections (using non-intersecting diagonals) of a convex (n+1)-gon into k-gons where k>=6. [_Joerg Arndt_, Feb 15 2014]
%H Vaclav Kotesovec, <a href="/A215342/b215342.txt">Table of n, a(n) for n = 1..250</a>
%H Alison Schuetz, Gwyneth Whieldon, <a href="http://arxiv.org/abs/1401.7194">Polygonal Dissections and Reversions of Series</a>, arXiv:1401.7194 [math.CO]
%H D. Birmajer, J. B. Gil, M. D. Weiner, <a href="https://arxiv.org/abs/1503.05242">Colored partitions of a convex polygon by noncrossing diagonals</a>, arXiv preprint arXiv:1503.05242 [math.CO], 2015.
%F Recurrence: 283*(n-3)*(n-2)*(n-1)*n*(14438110231*n^6 - 346214993274*n^5 + 3438949212625*n^4 - 18105364836570*n^3 + 53265099505324*n^2 - 82987438028496*n + 53465930027280)*a(n) = (n-3)*(n-2)*(n-1)*(5399853226394*n^7 - 137584187324067*n^6 + 1483504918415939*n^5 - 8763694066910355*n^4 + 30585682233578711*n^3 - 62946796681030518*n^2 + 70573456271906136*n - 33158656683118080)*a(n-1) - (n-3)*(n-2)*(534210078547*n^8 - 14946795065326*n^7 + 180235890644998*n^6 - 1221993860476624*n^5 + 5087726442447403*n^4 - 13295664719568394*n^3 + 21246368278875372*n^2 - 18919520411340456*n + 7154560952974080)*a(n-2) - 5*(n-4)*(n-3)*(28876220462*n^8 - 793496758165*n^7 + 9354947999333*n^6 - 61627542806839*n^5 + 247116200695877*n^4 - 613894185501244*n^3 + 913857055091496*n^2 - 732955177968120*n + 234541607788800)*a(n-3) + 5*(9947857949159*n^10 - 357916425755694*n^9 + 5753280746412201*n^8 - 54388490463504720*n^7 + 334719732595671573*n^6 - 1400557913088383070*n^5 + 4032929135663406319*n^4 - 7886242788829977540*n^3 + 10015113186875731788*n^2 - 7452248915385205056*n + 2464721951024954880)*a(n-4) - 8*(n-5)*(2*n - 9)*(4*n - 21)*(4*n - 19)*(14438110231*n^6 - 259586331888*n^5 + 1924445899720*n^4 - 7522955714190*n^3 + 16337121992089*n^2 - 18661982982042*n + 8745398997120)*a(n-5). - _Vaclav Kotesovec_, Mar 22 2014
%F a(n) ~ s*sqrt((3*r*s+r-4)/(5*(3*r*s-2*r-2))) / (2*sqrt(Pi) * n^(3/2) * r^(n-1)), where s = 1.1954869989505368389... is the root of the equation 64 - 897*s + 2460*s^2 - 2787*s^3 + 1442*s^4 - 283*s^5 = 0, and r = (4*s-5)/(s*(3*s-4)) = 0.441061092405258554919... - _Vaclav Kotesovec_, Mar 23 2014
%F G.f. A(x) for offset 0 satisfies 1-(1+x)*A(x) + x*A(x)^2 + x^4*A(x)^5 = 0. - _Vaclav Kotesovec_, Mar 23 2014
%e Use the Lang and the Abramowitz and Stegun links in A111785. In the A-S list of partitions of the integer n on page 831 null all partitions containing 1, 2, or 3. These correspond to the null coefficients of x^2, x^3, and x^4 in the series to be reverted and to 3-, 4-, and 5-gons not being allowed in the dissections. a(9)=6 corresponds to the A-S partitions (n=8,m=1, partition 1)=8 and (8,2,4)=4^2, and these in turn correspond to one undissected 10-gon + five ways to divide a 10-gon into two 6-gons. a(10)=12 corresponds to (9,1,1)=9 and (9,2,4)=4,5, corresponding to one undissected 11-gon + the eleven ways to divide an 11-gon into a 6-gon and 7-gon. - _Tom Copeland_, Feb 15 2014
%t nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=0; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[1-(1+x)*AGF+x*AGF^2 +x^4*AGF^5,x,j]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* _Vaclav Kotesovec_, Mar 23 2014 *)
%o (PARI) N=66; Vec(serreverse(x*(1-x^3*sum(k=1,N,x^k))+O(x^N)))
%Y Cf. A001003 (rev. of x*(1-1*sum(k>=1,x^k)) ), A046736 (rev. of x*(1-x*sum(k>=1,x^k)) ), A054514 (rev. of x*(1-x^2*sum(k>=1,x^k)) ).
%Y Cf. A000108 (rev. of x/(1+1*sum(k>=1,x^k)) ), A005043 (rev. of x/(1+x*sum(k>=1,x^k)) ), A114997 (rev. of x/(1+x^2*sum(k>=1,x^k)) ), A215341 (rev. of x/(1+x^3*sum(k>=1,x^k)) ).
%K nonn
%O 1,9
%A _Joerg Arndt_, Aug 19 2012
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