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%I #28 May 07 2019 10:59:34
%S 1,1,2,1,6,7,3,24,46,40,15,120,326,430,315,105,720,2556,4536,4900,
%T 3150,945,5040,22212,49644,70588,66150,38115,10395,40320,212976,
%U 574848,1011500,1235080,1032570,540540,135135
%N Triangle read by rows: labeled trees counted by improper edges.
%C 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.
%H J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.2010">Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure</a>, arXiv:1307.2010 [math.CO], 2013-2014.
%H Dominique Dumont, Armand Ramamonjisoa, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i2r17">Grammaire de Ramanujan et Arbres de Cayley</a>, Electr. J. Combinatorics, Volume 3, Issue 2 (1996) R17 (see page 17).
%H M. Josuat-Vergès, <a href="http://arxiv.org/abs/1310.7531">Derivatives of the tree function</a>, arXiv preprint arXiv:1310.7531 [math.CO], 2013.
%H Lucas Randazzo, <a href="https://arxiv.org/abs/1905.02083">Arboretum for a generalization of Ramanujan polynomials</a>, arXiv:1905.02083 [math.CO], 2019.
%H Jiang Zeng, <a href="http://math.univ-lyon1.fr/homes-www/zeng/public_html/paper/publication.html">A Ramanujan sequence that refines the Cayley formula for trees</a>, Ramanujan Journal 3 (1999) 1, 45-54, <a href="http://dx.doi.org/10.1023/A:1009809224933">[DOI]</a>
%e Table begins
%e \ k 0....1....2....3 ...
%e n
%e 1 |..1
%e 2 |..1
%e 3 |..2....1
%e 4 |..6....7....3
%e 5 |.24...46...40....15
%e 6 |120..326..430...315...105
%e 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.
%t 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]];
%t Join[{1}, Table[T[n,k], {n, 12}, {k, 0, n-2}]//Flatten] (* modified by _G. C. Greubel_, May 07 2019 *)
%o (Sage)
%o def T(n, k):
%o if k==0: return factorial(n-1)
%o elif (k<0 or k > n-2): return 0
%o else: return (n-1)*T(n-1, k) + (n+k-3)* T(n-1, k-1)
%o [1] + [[T(n, k) for k in (0..n-2)] for n in (2..12)] # _G. C. Greubel_, May 07 2019
%Y Cf. A054589, A075856. Row sums are n^(n-2), A000272.
%K nonn,tabf
%O 1,3
%A _David Callan_, Oct 14 2012
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