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A000309
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Number of rooted planar bridgeless cubic maps with 2n nodes.
(Formerly M3601 N1460)
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17
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1, 1, 4, 24, 176, 1456, 13056, 124032, 1230592, 12629760, 133186560, 1436098560, 15774990336, 176028860416, 1990947110912, 22783499599872, 263411369705472, 3073132646563840, 36143187370967040, 428157758086840320, 5105072641718353920, 61228492804372561920
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OFFSET
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0,3
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COMMENTS
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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
Description trees of type (2,2) with n edges. (A description tree of type (a,b) is a rooted plane tree where every internal node is labeled by an integer between a and [b + sum of labels of its children], every leaf is labeled a, and the root is labeled [b + sum of labels of its children]. See Definition 1 and Section 5.2 of Cori and Schaeffer 2003.) - Noam Zeilberger, Oct 08 2017
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018
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REFERENCES
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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).
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LINKS
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FORMULA
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a(n) = 2^(n-1) * A000139(n) for n > 0.
a(n) = 4*a(n-1)*binomial(3*n, 3) / binomial(2*n+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
D-finite with recurrence (n+1)*(2*n+1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Nov 02 2018
a(n) = -(-2)^(n-1)*(3*n+2)*hypergeom([-3*(n+1),-n,-n+1/3], [-n-1,-n-2/3], 1). The a(n) are values of the polynomials A358091. - Peter Luschny, Oct 28 2022
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MAPLE
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a := n -> 2^(n+1)*(3*n)!/(n!*(2*n+2)!);
A000309 := n -> -(-2)^(n-1)*(3*n+2)*hypergeom([-3*(n+1), -n, -n+1/3], [-n-1, -n-2/3], 1): seq(simplify(A000309(n)), n = 0..21); # Peter Luschny, Oct 28 2022
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MATHEMATICA
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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 *)
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PROG
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(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
(Sage) [2^n*factorial(3*n)/(factorial(n+1)*factorial(2*n+1))for n in range(20)] # G. C. Greubel Nov 29 2018
(GAP) List([0..20], n -> 2^(n+1)*Factorial(3*n)/(Factorial(n)* Factorial(2*n+2))); # G. C. Greubel, Nov 29 2018
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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