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A006963 Number of planar embedded labeled trees with n nodes: (2n-3)!/(n-1)! for n >= 2, a(1) = 1.
(Formerly M3076)
19
1, 1, 3, 20, 210, 3024, 55440, 1235520, 32432400, 980179200, 33522128640, 1279935820800, 53970627110400, 2490952020480000, 124903451312640000, 6761440164390912000, 393008709555221760000, 24412776311194951680000, 1613955767240110694400000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For n>1: central terms of the triangle in A173333; cf. A001761, A001813. - Reinhard Zumkeller, Feb 19 2010

Can be obtained from the Vandermonde permanent of the first n positive integers; see A093883. - Clark Kimberling, Jan 02 2012

All trees can be embedded in the plane, but "planar embedded" means that orientation matters but rotation doesn't. For example, the n-star with n-1 edges has n! ways to label it, but rotation removes a factor of n-1. Another example, the n-path has n! ways to label it, but rotation removes a factor of 2. - Michael Somos, Aug 19 2014

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

David Callan, A quick count of plane (or planar embedded) labeled trees

R. Coquereaux, J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: http://dx.doi.org/10.1142/S0218216516500474

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 109

B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.

P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992.

P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)

J. W. Moon, Counting Labelled Trees, Issue 1 of Canadian mathematical monographs, Canadian Mathematical Congress, 1970.

Ran J. Tessler, A Cayley-type identity for trees, arXiv:1809.00001 [math.CO], 2018.

Index entries for sequences related to trees

FORMULA

E.g.f. for a(n+1), n >= 1, log(c(x)); c(x) = g.f. for Catalan numbers A000108. - Wolfdieter Lang

Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n)=int(x^n*erfc(sqrt(x)/2)/2, x=0..infinity), n=0, 1..., where erfc(x) is the complementary error function. - Karol A. Penson, Sep 27 2001

a(n) ~ 2^(-5/2)*n^-2*2^(2*n)*e^-n*n^n - Joe Keane (jgk(AT)jgk.org), Jun 06 2002

a(n+1) = (n+1)*(n+2)*...*(2n-1) for n>=2. - Jaroslav Krizek, Nov 09 2010

E.g.f. (A(x)-1) is reversion of =exp(-x)-exp(-2*x). - Vladimir Kruchinin, Jan 30 2012

G.f.: 1 + x*G(0) where G(k) = 1 + x*(2*k+1)*(4*k+3)/(k + 1 - 4*x*(k+1)^2*(4*k+5)/(4*x*(k+1)*(4*k+5) + (2*k+3)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 02 2013

E.g.f.: 1 + x*E(0) where E(k) = 1 + x*(2*k+1)*(4*k+3)/(2*(k + 1)^2 - 8*x*(k+1)^3*(4*k+5)/(4*x*(k+1)*(4*k+5) + (2*k+3)^2/E(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 02 2013

E.g.f: sqrt(1-4*x)/4 - 1/4 + 3*x/2 - x*log((1+sqrt(1-4*x))/2). - Robert Israel, Aug 20 2014

(-n+1)*a(n) +2*(2*n-3)*(n-2)*a(n-1)=0. - R. J. Mathar, Jan 03 2018

EXAMPLE

G.f = x + x^2 + 3*x^3 + 20*x^4 + 210*x^5 + 3024*x^6 + 55440*x^7 + 1235520*x^8 + ...

a(5) = 210 = 30 + 60 + 120 where 30 is for the star, 60 for the path, and 120 for the tree with one trivalent vertex. - Michael Somos, Aug 19 2014

MAPLE

1, seq((2*n-3)!/(n-1)!, n=2..30); # Robert Israel, Aug 20 2014

MATHEMATICA

Join[{1}, Table[(2n-3)!/(n-1)!, {n, 2, 20}]] (* Harvey P. Dale, Nov 03 2011 *)

a[ n_] := With[{m = n - 1}, If[m < 1, Boole[m == 0], m! SeriesCoefficient[ -Log[(1 + Sqrt[1 - 4 x]) / 2], {x, 0, m}]]] (* Michael Somos, Jul 01 2013 *)

a[ n_] := If[n < 2, Boole[n == 1], (2 n - 3)! / (n - 1)!]; (* Michael Somos, Aug 19 2014 *)

PROG

(MAGMA) [1] cat [Factorial(2*n-3)/Factorial(n-1): n in [2..20]]; // Vincenzo Librandi, Nov 12 2011

(PARI) {a(n) = n--; if( n<1, n==0, n! * polcoeff( -log( (1 + sqrt(1 - 4*x + x * O(x^n))) / 2), n))}; /* Michael Somos, Jul 01 2013 */

CROSSREFS

Sequence in context: A218673 A230478 A014068 * A243426 A113333 A206405

Adjacent sequences:  A006960 A006961 A006962 * A006964 A006965 A006966

KEYWORD

nonn,easy,nice

AUTHOR

Simon Plouffe and N. J. A. Sloane

STATUS

approved

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Last modified December 17 00:41 EST 2018. Contains 318191 sequences. (Running on oeis4.)