

A054589


Table related to labeled rooted trees, cycles and binary trees.


6



1, 1, 1, 2, 4, 3, 6, 18, 25, 15, 24, 96, 190, 210, 105, 120, 600, 1526, 2380, 2205, 945, 720, 4320, 13356, 26488, 34650, 27720, 10395, 5040, 35280, 128052, 305620, 507430, 575190, 405405, 135135
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OFFSET

1,4


COMMENTS

The left column is (n1)!, the right column is (2n3)!!, the total of each line is n^{n1}.
Constant terms of polynomials related to Ramanujan psi polynomials (see Zeng reference).
From Peter Bala, Sep 29 2011: (Start)
Differentiating n times the Lambert function W(x) = sum {n>=1} n^(n1)*x^n/n! with respect to x yields (d/dx)^n W(x) = exp(n*W(x))/(1W(x))^n*R(n,1/(1W(x))), where R(n,x) is the nth row polynomial of this triangle. The first few values are R(1,x) = 1, R(2,x) = 1+x, R(3,x) = 2+4*x+3*x^2. The Ramanujan polynomials R(n,x) are strongly xlogconvex [Chen et al.].
Shor and DumontRamamonjisoa have proved independently that the coefficient of x^k in R(n,x) counts rooted labeled trees on n vertices with k improper edges. Drake, Example 1.7.3, gives another combinatorial interpretation for this triangle as counting a family of labeled trees.
(End)


LINKS

Table of n, a(n) for n=1..36.
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], 20132014.
W. Chen, L. X. W. Wang and A. L. B. Yang, Recurrence Relations for Strongly qLogConvex Polynomials, arXiv:0806.3641v1 [math.CO], 2008.
D. Dominici, Nested derivatives: A simple method for Computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.
B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
Dominique Dumont, Armand Ramamonjisoa, Grammaire de Ramanujan et Arbres de Cayley, Electr. J. Combinatorics, Volume 3, Issue 2 (1996) R17 (see page 16).
D. J. Jeffrey, G. A. Kalugin, N. Murdoch, Lagrange inversion and Lambert W, Preprint 2015.
M. JosuatVergès, Derivatives of the tree function, arXiv preprint arXiv:1310.7531 [math.CO], 2013.
Peter W. Shor, A new proof of Cayley's formula for counting labeled trees, J. Combin. Theory Ser. A 71 (1995), no. 1, 154158.
Jiang Zeng, A Ramanujan sequence that refines the Cayley formula for trees, Ramanujan J., 3(1999) 1, 4554.


FORMULA

The polynomials p_n=sum a[n, k]x^k satisfy p_1=1 and p_{n+1}=x*x*dp_n/dx+n*(1+x)*p_n
From Peter Bala, Sep 29 2011: (Start)
E.g.f.: series reversion with respect to x of (1t+(t1+x*t)*exp(x)) = x+(1+t)*x^2/2!+(2+4*t+3*t^2)*x^3/3!+....
The sequence of shifted row polynomials {p_n(1+t)}n>=1 begins [1,2+t,9+10*t+3*t^2,...]. These are the row polynomials of A048160.
(End)
Let f(x) = exp(x)/(1t*x). The e.g.f. A(x,t) = x+(1+t)*x^2/2!+(2+4*t+3*t^2)*x^3/3! + ... satisfies the autonomous differential equation dA/dx = f(A). The nth row polynomial (n>=1) equals D^(n1)(f(x)) evaluated at x = 0, where D is the operator f(x)*d/dx (apply [Dominici, Theorem 4.1]).  Peter Bala, Nov 09 2011
The polynomials (1+t)^(n1)*p_n(1/(1+t)) are (up to sign) the row polynomials of A042977.  Peter Bala, Jul 23 2012
Let q_n = sum(k>=0, a(n,k)*t^(nk) ), with q_0 = 1. (So q_1=t, q_2 = t+t^2, and q_3=3*t+4*t^2+2*t^3.) Then sum(n>=0, q_n*x^n/n! ) = t  W((t1t^2*x)*exp(t1)), where W is the Lambert function.  Ira M. Gessel, Jan 06 2012


EXAMPLE

{1}, {1, 1}, {2, 4, 3}, {6, 18, 25, 15}, etc.


MATHEMATICA

p[1] = 1; p[n_] := p[n] = Expand[x^2*D[p[n1], x] + (n1)(1+x)p[n1]]; Flatten[ Table[ CoefficientList[ p[n], x], {n, 1, 8}]] (* JeanFrançois Alcover, Jul 22 2011 *)
Clear[a];
a[1, 0] = 1;
a[n_, k_] /; k < 0  k >= n := 0
a[n_, k_] /; 0 <= k <= n  1 :=
a[n, k] = (n  1) a[n  1, k] + (n + k  2) a[n  1, k  1]
Table[a[n, k], {n, 20}, {k, 0, n  1} (* David Callan, Oct 14 2012 *)


CROSSREFS

Cf. A000169, A001147, A075856, A048159, A048160, A042977.
Sequence in context: A091274 A330745 A122525 * A051851 A336165 A011171
Adjacent sequences: A054586 A054587 A054588 * A054590 A054591 A054592


KEYWORD

nonn,tabl


AUTHOR

F. Chapoton, Apr 14 2000


STATUS

approved



