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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, 1, 57, 180, 122, 34, 192, 76, 15, 26, 16, 1, 1, 120, 768, 423, 496, 1494, 426, 294, 267, 474, 156, 56, 42, 22, 1, 1, 247, 2904, 1389, 4288, 9204, 2127, 496, 5946, 2829, 5142, 1206, 855, 768, 1344, 1038, 288, 56, 98, 64, 29, 1 (list; graph; refs; listen; history; text; internal format)
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 h-polynomials associated to the refined f-polynomials 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 Schröder'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 Schröder to Fatou to Julia, Friedrich Vieweg & Sohn, 1994

LINKS

Table of n, a(n) for n=0..66.

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

Tom Copeland, Mathemagical Forests v2, Posted June 2008

Tom Copeland, Addendum to Mathemagical Forests

P. Feinsilver, Lie algebras, representations, and analytic semigroups through dual vector fields, Cimpa-Unesco-Venezuela School, Mérida, Venezuela, Jan-Feb 2006.

P. Feinsilver, R. Schott, Appell systems on Lie groups, J. Theor. Probab. 5 (2) (1992) 251-281

V. Goryainov and O. Kudryavtseva One-parameter semigroups of analytic functions, fixed points and the Koenigs function , Sbornik: Mathematics, 202:7 (2011), 971-1000

J. Hoffman, Topics in Elliptic Curves and Modular Forms, p. 10.

C. Houzel, The Work of Niels Henrik Abel, The Legacy of Niels Henrik Abel-The Abel Bicentennial, Oslo 2002 (Editors O. Laudal and R Piene), Springer-Verlag (2004), pp. 24-25.

Peter Luschny, Expansion A145271 (added Jul 21 2016)

MathStackExchange, Closed form for sequence A145271

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

R^5 g(x) = 1 (0') (1')^5 + 26 (0')^2 (1')^3 (2') + (0')^3 [34 (1') (2')^2 + 32 (1')^2 (3')] + (0')^4 [ 15 (2') (3') + 11 (1') (4')] + (0')^5 (5')

R^6 g(x) = 1 (0') (1')^6 + 57 (0')^2 (1')^4 (2') + (0')^3 [180 (1')^2 (2')^2 + 122 (1')^3 (3')] + (0')^4 [ 34 (2')^3 + 192 (1') (2') (3') + 76 (1')^2 (4')] + (0')^5 [15 (3')^2 + 26 (2') (4') + 16 (1') (5')] + (0')^6 (6')

where (j')^k = ((d/dx)^j g(x))^k. And R^(n-1) g(x) evaluated at x=0 is the n-th 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 n-th diagonal (with n=0 the main diagonal) of the lower triangular Pascal matrix by g_n = (d/dx)^n g(x) to obtain the matrix VP with VP(n,k) = binomial(n,k) g_(n-k). Then R^n g(x) = (1, 0, 0, 0,..) [VP * S]^n (g_0, g_1, g_2, ..)^T, where S is the shift matrix A129185, representing differentiation in the divided powers basis x^n/n!. - Tom Copeland, Feb 10 2016 (An evaluation removed by author on Jul 19 2016. Cf. A139605 and A134685.)

Also, R^n g(x) = (1, 0, 0, 0,..) [VP * S]^(n+1) (0, 1, 0, ..)^T in agreement with A139605. - Tom Copeland, Jul 21 2016

EXAMPLE

From Tom Copeland, Sep 19 2014: (Start)

Let h(x)=log((1+a*x)/(1+b*x))/(a-b); 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)

Triangle starts:

1

1

1, 1

1, 4, 1

1, 11, 4, 7, 1

1, 26, 34, 32, 15, 11, 1

1, 57, 180, 122, 34, 192, 76, 15, 26, 16, 1

1, 120, 768, 423, 496, 1494, 426, 294, 267, 474, 156, 56, 42, 22, 1

1, 247, 2904, 1389, 4288, 9204, 2127, 496, 5946, 2829, 5142, 1206, 855, 768, 1344, 1038, 288, 56, 98, 64, 29, 1

MAPLE

with(LinearAlgebra): with(ListTools):

A145271_row := proc(n) local b, M, V, U, G, R, T;

if n < 2 then return 1 fi;

b := (n, k) -> `if`(k=1 or k>n+1, 0, binomial(n-1, k-2)*g[n-k+1]);

M := n -> Matrix(n, b):

V := n -> Vector[row]([1, seq(0, i=2..n)]):

U := n -> VectorMatrixMultiply(V(n), M(n)^(n-1)):

G := n -> Vector([seq(g[i], i=0..n-1)]);

R := n -> VectorMatrixMultiply(U(n), G(n)):

T := Reverse([op(sort(expand(R(n+1))))]);

seq(subs({seq(g[i]=1, i=0..n)}, T[j]), j=1..nops(T)) end:

for n from 0 to 9 do A145271_row(n) od; # Peter Luschny, Jul 20 2016

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).

Cf. A008292, A049019, A074060, A129185, A139002, A139605.

Second column is A000295, subdiagonal is A000124, row sums are A000142, row lengths are A000041. - Peter Luschny, Jul 21 2016

Sequence in context: A264614 A261762 A225062 * A232774 A203860 A147564

Adjacent sequences:  A145268 A145269 A145270 * A145272 A145273 A145274

KEYWORD

easy,nonn,tabf

AUTHOR

Tom Copeland, Oct 06 2008

EXTENSIONS

Title amplified by Tom Copeland, Mar 17 2014

R^5 and R^6 formulas and terms a(19)-a(29) added by Tom Copeland, Jul 11 2016

More terms from Peter Luschny, Jul 20 2016

STATUS

approved

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Last modified August 25 04:05 EDT 2016. Contains 275791 sequences.