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A002712 Number of unrooted triangulations of a disk that have reflection symmetry with n interior nodes and 3 nodes on the boundary.

%I A002712 M2746 N1103 #34 Apr 25 2023 07:57:59
%S A002712 1,1,1,3,8,23,68,215,680,2226,7327,24607,83060,284046,975950,3383343,
%T A002712 11778308,41269252,145131502,512881550,1818259952,6470758289,
%U A002712 23091680690,82659905947,296605398856,1067012168350,3846553544904,13896522968160,50296815014780,182378110257354,662384549806938
%N A002712 Number of unrooted triangulations of a disk that have reflection symmetry with n interior nodes and 3 nodes on the boundary.
%C A002712 These are also called [n,0]-triangulations.
%D A002712 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002712 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002712 Andrew Howroyd, <a href="/A002712/b002712.txt">Table of n, a(n) for n = 0..500</a>
%H A002712 William G. Brown, <a href="http://dx.doi.org/10.1112/plms/s3-14.4.746">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
%H A002712 W. G. Brown, <a href="/A002709/a002709.pdf">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]
%H A002712 C. F. Earl and L. J. March, <a href="/A005500/a005500_1.pdf">Architectural applications of graph theory</a>, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
%p A002712 Dc := proc(n,m) 2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)! ; end:
%p A002712 A000260 := proc(n) Dc(n,0) ; end:
%p A002712 Dx2 := proc(nmax) add( A000260(n)*x^(2*n),n=0..nmax) ; end:
%p A002712 o := 20: Order := 2*o-1 : j := solve( J0=1+x*J0+x^2*J0*(1+x*J0/2)*series(J0^2-Dx2(o),x=0,2*o-1),J0) ;
%p A002712 for n from 0 to 2*o-2 do printf("%d,",coeftayl(j,x=0,n)) ; od: # _R. J. Mathar_, Oct 29 2008
%t A002712 seq[m_] := Module[{q}, q = Sum[x^(2n) Binomial[4n+2, n+1]/ ((2n+1)(3n+2)), {n, 0, Quotient[m, 2]}]; p = 1+O[x]; Do[p = 1 + x*p + x^2*p*(1+x*p/2)(p^2-q), {n, 1, m}]; CoefficientList[p, x]];
%t A002712 seq[30] (* _Jean-François Alcover_, Apr 25 2023, after _Andrew Howroyd_ *)
%o A002712 (PARI) seq(n)={my(q=sum(n=0, n\2, x^(2*n)*binomial(4*n+2, n+1)/((2*n+1)*(3*n+2))), p=1+O(x)); for(n=1, n, p = 1 + x*p + x^2*p*(1 + x*p/2)*(p^2 - q)); Vec(p)} \\ _Andrew Howroyd_, Feb 24 2021
%Y A002712 Column k=0 of A169809.
%K A002712 nonn
%O A002712 0,4
%A A002712 _N. J. A. Sloane_
%E A002712 More terms from _R. J. Mathar_, Oct 29 2008
%E A002712 Name clarified and terms a(27) and beyond from _Andrew Howroyd_, Feb 24 2021