A038058 -id:A038058 - cp's OEIS frontend

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

Ralf Stephan, Aug 17 2004

nonn

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.

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.

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

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

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 *)

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

/* 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),", "))

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

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

A249632 Triangular array read by rows. T(n,k) is the number of labeled trees with black and white nodes having exactly k black nodes, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 9, 9, 3, 16, 64, 96, 64, 16, 125, 625, 1250, 1250, 625, 125, 1296, 7776, 19440, 25920, 19440, 7776, 1296, 16807, 117649, 352947, 588245, 588245, 352947, 117649, 16807, 262144, 2097152, 7340032, 14680064, 18350080, 14680064, 7340032, 2097152, 262144

Offset: 0

Geoffrey Critzer, Nov 02 2014

Row sums = A038058.
T(n,n) = T(n,0) = n^(n-2) free trees A000272.
T(n,n-1) = T(n,1) = n^(n-1) rooted trees A000169.
T(n,2) = A081131.

			1,
1,    1,
1,    2,    1,
3,    9,    9,     3,
16,   64,   96,    64,    16,
125,  625,  1250,  1250,  625,   125,
1296, 7776, 19440, 25920, 19440, 7776, 1296

F. Harary and E. Palmer, Graphical Enumeration, Academic Press,1973, page 30, exercise 1.10.

nn = 6; f[x_] := Sum[n^(n - 2) x^n/n!, {n, 1, nn}];
Map[Select[#, # > 0 &] &,
  Range[0, nn]! CoefficientList[
    Series[f[x + y x] + 1, {x, 0, nn}], {x, y}]] // Grid

E.g.f.: A(x + y*x) where A(x) is the e.g.f. for A000272.

cp's OEIS Frontend

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

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