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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
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
Sequence in context: A179015 A195621 A267090 * A142148 A142583 A327838
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.)