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
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
KEYWORD
nonn,tabf
AUTHOR
David Callan, Oct 14 2012
STATUS
approved