A097629 a(n) = 2*(2n)^(n-2).

1, 2, 12, 128, 2000, 41472, 1075648, 33554432, 1224440064, 51200000000, 2414538435584, 126806761930752, 7340688973975552, 464436530178424832, 31886460000000000000, 2361183241434822606848, 187591757103747287810048

Offset: 1

Author

Ralf Stephan, Aug 17 2004

Keywords

• nonn

Comments

Number of all unrooted directed trees on n nodes.

Ditrees are well-colored directed trees. Well-colored means, each green vertex has at least a red child, each red vertex has no red child.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..352
• Federico Ardila, Matthias Beck, Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
• C. Banderier, J.-M. Le Bars and V. Ravelomanana, Generating functions for kernels of digraphs, arXiv:math/0411138 [math.CO], 2004.
• Vsevolod Gubarev, Rota-Baxter operators on a sum of fields, arXiv:1811.08219 [math.RA], 2018.
• Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions (2008); arXiv:0806.3682 [math.CO], 2008. Discrete Math. 310 (2010), no. 24, 3584-3606.
• Jean-Christophe Novelli and Jean-Yves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959 [math.CO], 2012.
• J.-B. Priez, A. Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv:1411.4161 [math.CO], 2014-2015.

Crossrefs

Equals (1/2) A038058 = A097630(n) + A097631(n). Cf. A052746, A097627.

Programs

Magma

[1] cat [2*(2*n)^(n-2): n in [2..20]]; // Vincenzo Librandi, Nov 19 2014

Mathematica

Table[2*(2*n)^(n - 2), {n, 1, 50}] (* or *) With[{nmax = 40}, CoefficientList[Series[-LambertW[-2*x]*(1+LambertW[-2*x]/2)/2, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 15 2017 *)

Maxima

a(n):=sum(k!*stirling2(n-1,k)*binomial(2*n,k),k,0,n-1)/(n); /* Vladimir Kruchinin, Nov 19 2014 */

PARI

/* E.g.f. when offset=0 satisfies: */
{a(n)=local(A=1+2*x);for(i=1,21,A=1+2*sum(n=1,21,x^(2*n-1)/(2*n-1)!*A^((4*n-1)/2))+x*O(x^n));n!*polcoeff(A,n)} \\ Paul D. Hanna, Sep 07 2012
for(n=0,20,print1(a(n),", "))

PARI

x='x+O('x^50); Vec(serlaplace(-lambertw(-2*x)*(1 + lambertw(-2*x)/2)/2)) \\ G. C. Greubel, Nov 15 2017

Formula

E.g.f.: A(x) = B(x)-B(x)^2, B(x) = e.g.f. of A052746 or A(x) = C(2*x)/2, C(x) = e.g.f. of A000272.

E.g.f. satisfies: A(x) = 1 + 2*Sum_{n>=1} x^(2*n-1)/(2*n-1)! * A(x)^((4*n-1)/2) when offset=0: A(x) = Sum_{n>=0} a(n)*x^n/n!. - Paul D. Hanna, Sep 07 2012

E.g.f. satisfies: A(x) = 1/A(-x*A(x)^2) when offset=0. - Paul D. Hanna, Sep 07 2012

a(n) = sum(k=0..n-1, k!*stirling2(n-1,k)*binomial(2*n,k))/n. - Vladimir Kruchinin, Nov 19 2014

E.g.f.: -LambertW(-2*x)*(1+LambertW(-2*x)/2)/2. - Vaclav Kotesovec, Dec 08 2014