A002712 Number of unrooted triangulations of a disk that have reflection symmetry with n interior nodes and 3 nodes on the boundary.
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
Keywords
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)
Crossrefs
Column k=0 of A169809.
Programs
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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
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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 *) -
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
Extensions
More terms from R. J. Mathar, Oct 29 2008
Name clarified and terms a(27) and beyond from Andrew Howroyd, Feb 24 2021
Comments