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

1, 2, 12, 128, 2000, 41472, 1075648, 33554432, 1224440064, 51200000000, 2414538435584, 126806761930752, 7340688973975552, 464436530178424832, 31886460000000000000, 2361183241434822606848, 187591757103747287810048, 15917322219892801768783872

Offset

1,2

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, and Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020. See p. 10.

Cyril Banderier, Jean-Marie Le Bars, and Vlady 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.

Pamela E. Harris, J. Carlos Martínez Mori, and Alexander N. Wilson, Inducing Parking Spaces to the Hyperoctahedral Group, Galois J. Alg. 3(1) (2027), 26-50. See pp. 35 (Theorem 3.8), 47.

Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions arXiv:0806.3682 [math.CO], 2008. See also Disc. 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.

Jean-Baptiste Priez and Aladin Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv:1411.4161 [math.CO], 2014-2015.

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) = (1/n) * Sum_{k=0..n-1} k!*Stirling2(n-1,k)*binomial(2*n,k). - Vladimir Kruchinin, Nov 19 2014

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

a(n) = A052746(n)/n. - Alois P. Heinz, Jun 22 2026

Mathematica

Table[2*(2*n)^(n - 2), {n, 1, 50}] (* G. C. Greubel, Nov 15 2017 *)
(* Alternative: *)
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 *)

Prog

(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), ", "))
(Maxima) a(n):=sum(k!*stirling2(n-1, k)*binomial(2*n, k), k, 0, n-1)/(n); /* Vladimir Kruchinin, Nov 19 2014 */
(Magma) [1] cat [2*(2*n)^(n-2): n in [2..20]]; // Vincenzo Librandi, Nov 19 2014
(PARI) my(x='x+O('x^50)); Vec(serlaplace(-lambertw(-2*x)*(1 + lambertw(-2*x)/2)/2)) \\ G. C. Greubel, Nov 15 2017

Crossrefs

Equals (1/2) A038058 = A097630(n) + A097631(n).

Cf. A052746, A097627.

Sequence in context: A367374 A367853 A228608 * A259267 A014235 A098628

Adjacent sequences: A097626 A097627 A097628 * A097630 A097631 A097632

Keyword

nonn,easy

Author

Ralf Stephan, Aug 17 2004

Status

approved