A002712 Number of unrooted triangulations of a disk that have reflection symmetry with n interior nodes and 3 nodes on the boundary. (Formerly M2746 N1103)

1, 1, 1, 3, 8, 23, 68, 215, 680, 2226, 7327, 24607, 83060, 284046, 975950, 3383343, 11778308, 41269252, 145131502, 512881550, 1818259952, 6470758289, 23091680690, 82659905947, 296605398856, 1067012168350, 3846553544904, 13896522968160, 50296815014780, 182378110257354, 662384549806938

Offset

0,4

Comments

These are also called [n,0]-triangulations.

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.

W. G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]

C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)

Maple

Dc := proc(n, m) 2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)! ; end:
A000260 := proc(n) Dc(n, 0) ; end:
Dx2 := proc(nmax) add( A000260(n)*x^(2*n), n=0..nmax) ; end:
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) ;
for n from 0 to 2*o-2 do printf("%d, ", coeftayl(j, x=0, n)) ; od: # R. J. Mathar, Oct 29 2008

Mathematica

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]];
seq[30] (* Jean-François Alcover, Apr 25 2023, after Andrew Howroyd *)

Prog

(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

Crossrefs

Column k=0 of A169809.

Sequence in context: A057198 A025262 A056010 * A192679 A193418 A005960

Adjacent sequences: A002709 A002710 A002711 * A002713 A002714 A002715

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from R. J. Mathar, Oct 29 2008

Name clarified and terms a(27) and beyond from Andrew Howroyd, Feb 24 2021

Status

approved