A097629 - OEIS

%I #53 Sep 08 2022 08:45:14

%S 1,2,12,128,2000,41472,1075648,33554432,1224440064,51200000000,

%T 2414538435584,126806761930752,7340688973975552,464436530178424832,

%U 31886460000000000000,2361183241434822606848,187591757103747287810048

%N a(n) = 2*(2n)^(n-2).

%C Number of all unrooted directed trees on n nodes.

%C 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.

%H G. C. Greubel, <a href="/A097629/b097629.txt">Table of n, a(n) for n = 1..352</a>

%H Federico Ardila, Matthias Beck, Jodi McWhirter, <a href="https://arxiv.org/abs/2004.02952">The Arithmetic of Coxeter Permutahedra</a>, arXiv:2004.02952 [math.CO], 2020.

%H C. Banderier, J.-M. Le Bars and V. Ravelomanana, <a href="https://arxiv.org/abs/math/0411138">Generating functions for kernels of digraphs</a>, arXiv:math/0411138 [math.CO], 2004.

%H Vsevolod Gubarev, <a href="https://arxiv.org/abs/1811.08219">Rota-Baxter operators on a sum of fields</a>, arXiv:1811.08219 [math.RA], 2018.

%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="http://arxiv.org/abs/0806.3682">Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions</a> (2008); arXiv:0806.3682 [math.CO], 2008. Discrete Math. 310 (2010), no. 24, 3584-3606.

%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="http://arxiv.org/abs/1209.5959">Duplicial algebras and Lagrange inversion</a>, arXiv preprint arXiv:1209.5959 [math.CO], 2012.

%H J.-B. Priez, A. Virmaux, <a href="http://arxiv.org/abs/1411.4161">Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration</a>, arXiv:1411.4161 [math.CO], 2014-2015.

%F 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.

%F 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

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

%F a(n) = sum(k=0..n-1, k!*stirling2(n-1,k)*binomial(2*n,k))/n. - _Vladimir Kruchinin_, Nov 19 2014

%F E.g.f.: -LambertW(-2*x)*(1+LambertW(-2*x)/2)/2. - _Vaclav Kotesovec_, Dec 08 2014

%t 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 *)

%o (PARI) /* E.g.f. when offset=0 satisfies: */

%o {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

%o for(n=0,20,print1(a(n),", "))

%o (Maxima) a(n):=sum(k!*stirling2(n-1,k)*binomial(2*n,k),k,0,n-1)/(n); /* _Vladimir Kruchinin_, Nov 19 2014 */

%o (Magma) [1] cat [2*(2*n)^(n-2): n in [2..20]]; // _Vincenzo Librandi_, Nov 19 2014

%o (PARI) x='x+O('x^50); Vec(serlaplace(-lambertw(-2*x)*(1 + lambertw(-2*x)/2)/2)) \\ _G. C. Greubel_, Nov 15 2017

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

%K nonn

%O 1,2

%A _Ralf Stephan_, Aug 17 2004