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A061356 Triangle T(n,k) = labeled trees on n nodes with maximal node degree k (0 < k < n). 10
1, 2, 1, 9, 6, 1, 64, 48, 12, 1, 625, 500, 150, 20, 1, 7776, 6480, 2160, 360, 30, 1, 117649, 100842, 36015, 6860, 735, 42, 1, 2097152, 1835008, 688128, 143360, 17920, 1344, 56, 1, 43046721, 38263752, 14880348, 3306744, 459270, 40824, 2268, 72, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

This is a formula from Theorem F, vol. I, p. 81 (French edition) used in proving Theorem D.

If we let N = n+1, binomial(N-2, k-1)*(N-1)^(N-k-1) = binomial(n-1, k-1)*n^(n-k), so this sequence with offset 1,1 also gives the number of rooted forests of k trees over [n]. - Washington Bomfim, Jan 09 2008

Let S(n,k) be the signed triangle, S(n,k) = (-1)^(n-k)T(n,k), which starts 1, -2, 1, 9, -6, 1,..., then the inverse of S is the triangle of idempotent numbers A059298. - Peter Luschny, Mar 13 2009

With offset 1 also number of labeled multigraphs of k components, n nodes, and no cycles except one loop in each component. See link below to have a picture showing the bijection between rooted forests and multigraphs of this kind. (Note that there are no labels in the picture, but the bijection remains true if we label the nodes.) - Washington Bomfim, Sep 04 2010

With offset 1, T(n,k) is the number of forests of rooted trees on n nodes with exactly k (rooted) trees. - Geoffrey Critzer, Feb 10 2012

REFERENCES

L. Comtet, Analyse Combinatoire, P.U.F., Paris 1970. Volume 1, p 81.

L. Comtet, Advanced Combinatorics, Reidel, 1974.

J. W. Moon, Another proof of Cayley's formula for counting trees, A.M.M., 70 (1963) p846-7.

LINKS

Table of n, a(n) for n=2..46.

P. Bala, Diagonals of triangles with generating function exp(t*F(x))

W. Bomfim, Bijection between rooted forests and multigraphs without cycles except one loop in each connected component. [From Washington Bomfim, Sep 04 2010]

Jim Pitman, Coalescent Random Forests .

E. W. Weisstein, Abel Polynomial, From MathWorld - A Wolfram Web Resource.

Wikipedia, Lambert W function

J. Zeng, A Ramanujan sequence that refines the Cayley formula for trees, Ramanujan J., 3(1999) 1, 45-54.

FORMULA

T(n, k) = binomial(n-2, k-1)*(n-1)^(n-k-1).

E.g.f.: (-LambertW(-y)/y)^(x+1)/(1+LambertW(-y)). - Vladeta Jovovic

From Peter Bala, Sep 21 2012: (Start)

Let T(x) = sum {n >= 0} n^(n-1) x^n/n! denote the tree function of A000169. E.g.f.: F(x,t) := exp(t*T(x)) - 1 = -1 + {T(x)/x}^t = t*x + t*(2 + t)*x^2/2! + t*(9 + 6*t + t^2)*x^3/3! + ....

The compositional inverse with respect to x of 1/t*F(x,t) is the e.g.f. for a signed version of the row reverse of A028421.

The row generating polynomials are the Abel polynomials A(n,x) = x*(x+n)^(n-1) for n >= 1.

Define B(n,x) = x^n/(1+n*x)^(n+1) = (-1)^n*A(-n,-1/x) for n >= 1. The k-th column entries are the coefficients in the formal series expansion of x^k in terms of B(n,x). For example, Col. 1: x = B(1,x) + 2*B(2,x) + 9*B(3,x) + 64*B(4,x) + ..., Col. 2: x^2 = B(2,x) + 6*B(3,x) + 48*B(4,x) + 500*B(5,x) + ... Compare with A059297.

n-th row sum = A000272(n+1). Row reverse triangle is A139526.

The o.g.f.'s for the diagonals of the triangle are the rational functions R(n,x)/(1-x)^(2*n+1), where R(n,x) are the row polynomials of A155163. See below for examples.

(End)

EXAMPLE

1

2     1

9     6     1

64    48    12    1

625   500   150   20    1

7776  6480  2160  360   30    1

...

From Peter Bala, Sep 21 2012: (Start)

O.g.f.'s for the diagonals begin:

1/(1-x) = 1 + x + x^2 + x^3 + ...

2*x/(1-x)^3 = 2 + 6*x + 12*x^3 + ...

(9+3*x)/(1-x)^5 = 9 + 48*x + 150*x^2 + ...

The numerator polynomials are the row polynomials of A155163.

(End)

MATHEMATICA

nn = 7; t = Sum[n^(n - 1)  x^n/n!, {n, 1, nn}]; f[list_] := Select[list, # > 0 &]; Map[f, Drop[Range[0, nn]! CoefficientList[Series[Exp[y t], {x, 0, nn}], {x, y}], 1]] // Flatten  (* Geoffrey Critzer, Feb 10 2012 *)

PROG

(Maxima) create_list(binomial(n, k)*(n+1)^(n-k), n, 0, 20, k, 0, n); /* Emanuele Munarini, Apr 01 2014 */

CROSSREFS

Columns are A000169, A053506, A053507, A053508, A053509.

First diagonal is A002378.

Sum of lines gives A000272.

Cf. A028421, A059297, A139526 (row reverse), A155163, A202017.

Sequence in context: A141618 A061691 A235595 * A141028 A246323 A201685

Adjacent sequences:  A061353 A061354 A061355 * A061357 A061358 A061359

KEYWORD

easy,nonn,tabl

AUTHOR

Olivier Gérard, Jun 07 2001

STATUS

approved

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Last modified October 2 02:22 EDT 2014. Contains 247527 sequences.