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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 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 April 21 10:28 EDT 2021. Contains 343149 sequences. (Running on oeis4.)