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 A067948 Triangle of labeled rooted trees according to the number of increasing edges. 4
 1, 1, 1, 2, 5, 2, 6, 26, 26, 6, 24, 154, 269, 154, 24, 120, 1044, 2724, 2724, 1044, 120, 720, 8028, 28636, 42881, 28636, 8028, 720, 5040, 69264, 319024, 655248, 655248, 319024, 69264, 5040 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Each line is symmetric. The sum of each line is n^(n-1), A000169. The outer diagonal is (n-1)!, A000142. The next-to-last diagonal is A001705. LINKS Table of n, a(n) for n=1..36. Brian Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.7.2), A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University. Ira M. Gessel and Seunghyun Seo, A refinement of Cayley's formula for trees, Electronic J. Combin. 11, no. 2 (2004-6) (The Stanley Festschrift volume). A. M. Khidr and B. S. El-Desouky, A symmetric sum involving the Stirling numbers of the first kind, European J. Combin., 5 (1984), 51-54. FORMULA G.f. of row n: Sum_{k=0..n-1} T(n, k) x^k = Product_{i=1..n-1} (n - i + i*x). From Peter Bala, Sep 29 2011: (Start) E.g.f.: Compositional inverse of (exp(x) - exp(x*t))/((1 - t)*exp(x*(1 + t))) = x + (1 + t)*x^2/2! + (2 + 5*t + 2*t^2)*x^3/3! + ... Let f(x,t) = (1 - t)/(exp(-x) - t*exp(-x*t)) and let D be the operator f(x,t)*d/dx. Then the (n+1)-th row generating polynomial equals (D^n)(f(x,t)) evaluated at x = 0. See [Drake, example 1.7.2] for the combinatorial interpretation of this table in terms of labeled trees. (End) EXAMPLE Triangle starts: 1; 1, 1; 2, 5, 2; 6, 26, 26, 6; 24, 154, 269, 154, 24; ... From Bruno Berselli, Jan 12 2021: (Start) The rows of the triangle are the coefficients of the following polynomials: 1: 1; 2: 1*x+1; 3: (x+2)*(2*x+1) = 2*x^2 + 5*x + 2; 4: (x+3)*(2*x+2)*(3*x+1) = 6*x^3 + 26*x^2 + 26*x + 6; 5: (x+4)*(2*x+3)*(3*x+2)*(4*x+1) = 24*x^4 + 154*x^3 + 269*x^2 + 154*x + 24, etc. (End) MATHEMATICA L := CoefficientList[InverseSeries[Series[(Exp[-x y] + Sinh[x] - Cosh[x])/(1 - y), {x, 0, 8}]], {x}]; Table[CoefficientList[L, y][[n + 1]] n!, {n, 1, 8}] // Flatten (* Peter Luschny, Jun 23 2018 *) CROSSREFS Cf. A000142, A000169, A001705. Sequence in context: A179015 A195621 A267090 * A142148 A142583 A327838 Adjacent sequences: A067945 A067946 A067947 * A067949 A067950 A067951 KEYWORD nonn,tabl AUTHOR Cedric Chauve (chauve(AT)lacim.uqam.ca), Mar 19 2002 STATUS approved

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Last modified June 23 05:14 EDT 2024. Contains 373629 sequences. (Running on oeis4.)