

A145271


Coefficients for expansion of (g(x)d/dx)^n g(x); refined Eulerian numbers for calculating compositional inverse of h(x)= (d/dx)^(1) 1/g(x); iterated derivatives as infinitesimal generators of flows.


32



1, 1, 1, 1, 1, 4, 1, 1, 11, 4, 7, 1, 1, 26, 34, 32, 15, 11, 1
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OFFSET

0,6


COMMENTS

For more detail, including connections to Legendre transformations, rooted trees, A139605, A139002 and A074060, see Copeland link p. 9.
For connections to the hpolynomials associated to the refined fpolynomials of permutohedra see my comments in A008292 and A049019.
From Tom Copeland, Oct 14 2011: (Start)
Given analytic functions F(x) and FI(x) such that F(FI(x))=FI(F(x))=x about 0, i.e., they are compositional inverses of each other, then, with g(x)= 1/dFI(x)/dx, a flow function W(s,x) can be defined with the following relations:
W(s,x)= exp(s g(x)d/dx)x = F(s+FI(x)) <flow fct.>,
W(s,0)= F(s), <orbit of the flow>,
W(0,x)= x <identity property>,
dW(0,x)/ds = g(x) = F'[FI(x)], <infinitesimal generator>, implying
dW(0,F(x))/ds = g(F(x)) = F'(x), <autonomous diff. eqn.>, and
W(s,W(r,x)) = F(s+FI(F(r+FI(x)))) = F(s+r+FI(x)) = W(s+r,x) <group property>. (See MF link below.) (End)
dW(s,x)/ds  g(x)dW(s,x)/dx = 0, so (1,g(x)) are the components of a vector orthogonal to the gradient of W and, therefore, tangent to the contour of W, at (s,x) <tangency property>.  Tom Copeland, Oct 26 2011
Though A139605 contains A145271, the op. of A145271 contains that of A139605 in the sense that exp(s g(x)d/dx) w(x)= w(F(s+FI(x)))
= exp((exp(s g(x)d/dx)x)d/du)w(u) evaluated at u=0. This is reflected in the fact that the forest of rooted trees assoc. to (g(x)d/dx)^n, FOR_n, can be generated by removing the single trunk of the planted rooted trees of FOR_(n+1).  Tom Copeland, Nov 29 2011
Related to formal group laws for elliptic curves (see Hoffman).  Tom Copeland, Feb 24 2012
The functional equation W(s,x)=F(s+FI(x)), or a restriction of it, is sometimes called the Abel equation or Abel's functional equation (see Houzel and Wikipedia) and is related to SchrÃ¶der's functional equation and Koenigs functions for compositional iterates (Alexander, Goryainov and Kudryavtseva).  Tom Copeland, Apr 04 2012


REFERENCES

D. S. Alexander, A History of Complex Dynamics: From SchrÃ¶der to Fatou to Julia, Friedrich Vieweg & Sohn, 1994


LINKS

Table of n, a(n) for n=0..18.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
Tom Copeland, The Elliptic Lie Triad: KdV and Ricatti Equations, Infinigens, and Elliptic Genera
Tom Copeland, Flipping Functions with Permutohedra Posted Oct. 2008 [From Tom Copeland, Oct 08 2008]
Tom Copeland, Mathemagical Forests v2 Posted June 2008
T. Copeland, Addendum to Mathemagical Forests
P. Feinsilver, Lie algebras, representations, and analytic semigroups through dual vector fields.  Tom Copeland, Sep 28 2015
V. Goryainov and O. Kudryavtseva Oneparameter semigroups of analytic functions, fixed points and the Koenigs function Sbornik: Mathematics, 202:7 (2011), 9711000
J. Hoffman, Topics in Elliptic Curves and Modular Forms p. 10.
C. Houzel, The Work of Niels Henrik Abel The Legacy of Niels Henrik AbelThe Abel Bicentennial, Oslo 2002 (Editors O. Laudal and R Piene), SpringerVerlag (2004), pp. 2425.
Wikipedia, Abel equation


FORMULA

Let R = g(x)d/dx; then
R^0 g(x) = 1 (0')^1
R^1 g(x) = 1 (0')^1 (1')^1
R^2 g(x) = 1 (0')^1 (1')^2 + 1 (0')^2 (2')^1
R^3 g(x) = 1 (0')^1 (1')^3 + 4 (0')^2 (1')^1 (2')^1 + 1 (0')^3 (3')^1
R^4 g(x) = 1 (0')^1 (1')^4 + 11 (0')^2 (1')^2 (2')^1 + 4 (0')^3 (2')^2 + 7 (0')^3 (1')^1 (3')^1 + 1 (0')^4 (4')^1
where (j')^k = ((d/dx)^j g(x))^k. And R^(n1) g(x) evaluated at x=0 is the nth Taylor series coefficient of the compositional inverse of h(x)= (d/dx)^(1) 1/g(x), with the integral from 0 to x.
The partitions are in reverse order to those in Abramowitz and Stegun p. 831. Summing over coefficients with like powers of (0') gives A008292.
Confer A190015 for another way to compute numbers for the array for each partition.  Tom Copeland, Oct 17 2014
Equivalent matrix computation: Multiply the nth diagonal of the lower triangular Pascal matrix by g_n = (d/dx)^n g(x)_{x=0}, i.e., the nth Taylor series coefficient of g(x) about x=0 to obtain the matrix VP with VP(n,k) = binomial(n,k) g(nk). Then R^n g(x) _{x=0} = (1, 0, 0, 0,..) [VP * S]^n (g_0, g_1, g_2, ..)^T, where S is the shift matrix A129185, representing differentiation of an e.g.f., or Taylor series.  Tom Copeland, Feb 10 2016


EXAMPLE

From Tom Copeland, Sep 19 2014: (Start)
Let h(x)=log((1+a*x)/(1+b*x))/(ab); then, g(x)=1/(dh(x)/dx)=(1+ax)(1+bx), so (0')=1, (1')= a+b, (2')= 2ab, evaluated at x=0, and higher order derivatives of g(x) vanish. Therefore, evaluated at x=0,
R^0 g(x) = 1
R^1 g(x) = a+b
R^2 g(x) = (a+b)^2 + 2ab = a^2 + 4 ab + b^2
R^3 g(x) = (a+b)^3 + 4 (a+b)*2ab = a^3 + 11 a^2*b + 11 ab^2 + b^3
R^4 g(x) = (a+b)^4 + 11 (a+b)^2*2ab + 4 (2ab)^2
= a^4 + 26 a^3*b + 66 a^2*b^2 + 26 ab^3 + b^4,
etc., and these bivariate Eulerian polynomials (A008292) are the first few coefficients of h^(1)(x)=(e^(ax)e^(bx))/(a*e^(bx)b*e^(ax)), the inverse of h(x). (End)


CROSSREFS

Cf. (A133437, A086810, A181289) = (LIF, reduced LIF, associated g(x)), where LIF is a Lagrange inversion formula.
Cf. (A134264, A001263, A119900), (A134685, A134991, A019538), (A133932, A111999, A007318).
Sequence in context: A264614 A261762 A225062 * A232774 A203860 A147564
Adjacent sequences: A145268 A145269 A145270 * A145272 A145273 A145274


KEYWORD

easy,nonn,more,tabf


AUTHOR

Tom Copeland, Oct 06 2008, Oct 08 2008


EXTENSIONS

Title amplified by Tom Copeland, Mar 17 2014


STATUS

approved



