A067948 - OEIS

%I #50 Jun 26 2024 19:34:21

%S 1,1,1,2,5,2,6,26,26,6,24,154,269,154,24,120,1044,2724,2724,1044,120,

%T 720,8028,28636,42881,28636,8028,720,5040,69264,319024,655248,655248,

%U 319024,69264,5040,40320,663696,3793212,10095228,13861809,10095228,3793212,663696,40320

%N Triangle of labeled rooted trees according to the number of increasing edges.

%C Each line is symmetric.

%C The sum of each line is n^(n-1), A000169.

%C The outer diagonal is (n-1)!, A000142.

%C The next-to-last diagonal is A001705.

%H Alois P. Heinz, <a href="/A067948/b067948.txt">Rows n = 1..141, flattened</a>

%H Brian Drake, <a href="http://people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf">An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.7.2)</a>, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.

%H Ira M. Gessel and Seunghyun Seo, <a href="https://doi.org/10.37236/1884">A refinement of Cayley's formula for trees</a>, Electronic J. Combin. 11, no. 2 (2004-6) (The Stanley Festschrift volume).

%H A. M. Khidr and B. S. El-Desouky, <a href="http://dx.doi.org/10.1016/S0195-6698(84)80018-9">A symmetric sum involving the Stirling numbers of the first kind</a>, European J. Combin., 5 (1984), 51-54.

%F G.f. of row n: Sum_{k=0..n-1} T(n, k) x^k = Product_{i=1..n-1} (n - i + i*x).

%F From _Peter Bala_, Sep 29 2011: (Start)

%F 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! + ...

%F 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)

%e Triangle starts:

%e    1;

%e    1,   1;

%e    2,   5,   2;

%e    6,  26,  26,   6;

%e   24, 154, 269, 154,  24;

%e   ...

%e From _Bruno Berselli_, Jan 12 2021: (Start)

%e The rows of the triangle are the coefficients of the following polynomials:

%e 1: 1;

%e 2: 1*x+1;

%e 3: (x+2)*(2*x+1) = 2*x^2 + 5*x + 2;

%e 4: (x+3)*(2*x+2)*(3*x+1) = 6*x^3 + 26*x^2 + 26*x + 6;

%e 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.

%e (End)

%p b:= proc(n) option remember;

%p       expand(x*mul(n-k+k*x, k=1..n-1))

%p     end:

%p T:= (n, k)-> coeff(b(n), x, k):

%p seq(seq(T(n,k), k=1..n), n=1..10);  # _Alois P. Heinz_, Jun 26 2024

%t 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 *)

%Y Cf. A000142, A000169, A001705.

%K nonn,tabl

%O 1,4

%A Cedric Chauve (chauve(AT)lacim.uqam.ca), Mar 19 2002