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A000309 Number of rooted planar bridgeless cubic maps with 2n nodes.
(Formerly M3601 N1460)
9
1, 1, 4, 24, 176, 1456, 13056, 124032, 1230592, 12629760, 133186560, 1436098560, 15774990336, 176028860416, 1990947110912, 22783499599872, 263411369705472, 3073132646563840, 36143187370967040, 428157758086840320 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also counts rooted planar non-separable triangulations with 3n edges. - Valery A. Liskovets, Dec 01 2003

Equivalently, rooted planar loopless triangulations with 2n triangles. - Noam Zeilberger, Oct 06 2016

REFERENCES

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.

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

T. D. Noe, Table of n, a(n) for n=0..100

Marie Albenque, Dominique Poulalhon, A Generic Method for Bijections between Blossoming Trees and Planar Maps, Electron. J. Combin., 22 (2015), #P2.38.

Olivier Bernardi, Bijective counting of Kreweras walks and loopless triangulations, Journal of Combinatorial Theory, Series A 114:5 (2007), 931-956.

Junliang Cai, Yanpei Liu, The enumeration of rooted nonseparable nearly cubic maps, Discrete Math. 207 (1999), no. 1-3, 9--24. MR1710479 (2000g:05074). See (31).

S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.

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)

R. C. Mullin, On counting rooted triangular maps, Canad. J. Math., v.17 (1965), 373-382.

W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.

W. T. Tutte, On the enumeration of four-colored maps, SIAM J. Appl. Math., 17 (1969), 454-460.

FORMULA

a(n) = 2^(n-1) * A000139(n) for n>0.

a(n) = 4*a(n-1)*binomial(3n, 3) / binomial(2n+2, 3); a(n) = 2^n*(3*n)!/ ( (n+1)!*(2*n+1)! ).

G.f.: (1/(6*x)) * (hypergeom([ -2/3, -1/3],[1/2],(27/2)*x)-1). [Mark van Hoeij, Nov 02 2009]

a(n) ~ 3^(3*n+1/2)/(sqrt(Pi)*2^(n+2)*n^(5/2)). - Ilya Gutkovskiy, Oct 06 2016

MAPLE

f:=n->2^(n+1)*(3*n)!/(n!*(2*n+2)!);

MATHEMATICA

f[n_] := 2^n(3n)!/((n + 1)!(2n + 1)!); Table[f[n], {n, 0, 19}] (* Robert G. Wilson v, Sep 21 2004 *)

Join[{1}, RecurrenceTable[{a[1]==1, a[n]==4a[n-1] Binomial[3n, 3]/ Binomial[2n+2, 3]}, a[n], {n, 20}]] (* Harvey P. Dale, May 11 2011 *)

PROG

(PARI) a(n) = 2^(n+1)*(3*n)!/(n!*(2*n+2)!); \\ Michel Marcus, Aug 09 2014

(MAGMA) [2^(n+1)*Factorial(3*n)/(Factorial(n)*Factorial(2*n+2)): n in [0..20]]; // Vincenzo Librandi, Aug 10 2014

CROSSREFS

Cf. A000139, A006335, A000264, A000356, A002005.

Sequence in context: A215709 A103334 A156017 * A112914 A007846 A139702

Adjacent sequences:  A000306 A000307 A000308 * A000310 A000311 A000312

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane and Robert G. Wilson v

EXTENSIONS

Definition clarified by Michael Albert, Oct 24 2008

STATUS

approved

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Last modified August 21 16:05 EDT 2017. Contains 290890 sequences.