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A217922 Triangle read by rows: labeled trees counted by improper edges. 0
1, 1, 2, 1, 6, 7, 3, 24, 46, 40, 15, 120, 326, 430, 315, 105, 720, 2556, 4536, 4900, 3150, 945, 5040, 22212, 49644, 70588, 66150, 38115, 10395, 40320, 212976, 574848, 1011500, 1235080, 1032570, 540540, 135135 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

T(n,k) is the number of labeled trees on [n], rooted at 1, with k improper edges, for n >= 1, k >= 0. See Zeng link for definition of improper edge.

LINKS

Table of n, a(n) for n=1..37.

J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013-2014.

Dominique Dumont, Armand Ramamonjisoa, Grammaire de Ramanujan et Arbres de Cayley, Electr. J. Combinatorics, Volume 3, Issue 2 (1996) R17 (see page 17).

M. Josuat-Vergès, Derivatives of the tree function, arXiv preprint arXiv:1310.7531 [math.CO], 2013.

Lucas Randazzo, Arboretum for a generalization of Ramanujan polynomials, arXiv:1905.02083 [math.CO], 2019.

Jiang Zeng, A Ramanujan sequence that refines the Cayley formula for trees, Ramanujan Journal 3 (1999) 1, 45-54, [DOI]

EXAMPLE

Table begins

\ k  0....1....2....3   ...

n

1 |..1

2 |..1

3 |..2....1

4 |..6....7....3

5 |.24...46...40....15

6 |120..326..430...315...105

T(4,2) = 3 because we have 1->3->4->2, 1->4->2->3, 1->4->3->2, in each of which the last 2 edges are improper.

MATHEMATICA

T[n_, 0]:= (n-1)!; T[n_, k_]:= If[k<0 || k>n-2, 0, (n-1)T[n-1, k] +(n+k-3)T[n-1, k-1]];

Join[{1}, Table[T[n, k], {n, 12}, {k, 0, n-2}]//Flatten] (* modified by G. C. Greubel, May 07 2019 *)

PROG

(Sage)

def T(n, k):

    if k==0: return factorial(n-1)

    elif (k<0 or k > n-2): return 0

    else: return (n-1)*T(n-1, k) + (n+k-3)* T(n-1, k-1)

[1] + [[T(n, k) for k in (0..n-2)] for n in (2..12)] # G. C. Greubel, May 07 2019

CROSSREFS

Cf. A054589, A075856. Row sums are n^(n-2), A000272.

Sequence in context: A192329 A059364 A258870 * A196554 A244647 A324037

Adjacent sequences:  A217919 A217920 A217921 * A217923 A217924 A217925

KEYWORD

nonn,tabf

AUTHOR

David Callan, Oct 14 2012

STATUS

approved

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Last modified November 12 14:43 EST 2019. Contains 329058 sequences. (Running on oeis4.)