OFFSET
0,6
COMMENTS
For more detail, including connections to Legendre transformations, rooted trees, A139605, A139002 and A074060, see Mathemagical Forests 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
g(W(s,x)) = F'(s + FI(x)) = dW(s,x)/ds = g(x) dW(s,x)/dx, connecting the operators here to presentations of the Koenigs / Königs function and Loewner / Löwner evolution equations of the Contreras et al. papers. - Tom Copeland, Jun 03 2018
The autonomous differential equation above also appears with a change in variable of the form x = log(u) in the renormalization group equation, or Beta function. See Wikipedia, Zinn-Justin equations 2.10 and 3.11, and Krajewski and Martinetti equation 21. - Tom Copeland, Jul 23 2020
A variant of these partition polynomials appears on p. 83 of Petreolle et al. with the indeterminates e_n there related to those given in the examples below by e_n = n!*(n'). The coefficients are interpreted as enumerating certain types of trees. See also A190015. - Tom Copeland, Oct 03 2022
REFERENCES
D. S. Alexander, A History of Complex Dynamics: From Schröder to Fatou to Julia, Friedrich Vieweg & Sohn, 1994.
T. Mansour and M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, Chapman and Hall/CRC, 2015.
LINKS
Luc Rousseau, Table of n, a(n) for n = 0..9295 (Rows 0 to 25, flattened).
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.
F. Bracci, M. Contreras, and S. Díaz-Madrigal, On the Königs function of semigroups of holomorphic self-maps of the unit disc, arXiv:1804.10465 [math.CV], p. 15, 2018.
F. Bracci, M. Contreras, S. Díaz-Madrigal, and A. Vasil'ev, Classical and stochastic Löwner-Kufarev equations , arXiv:1309.6423 [math.CV], (cf., e.g., p. 23), 2013.
C. Brouder, Trees, renormalization, and differential equations, BIT Numerical Mathematics, 44: 425-438, 2004, (Flow / autonomous differential equation, p. 429).
M. Contreras, S. Díaz-Madrigal, and P. Gumenyuk, Loewner chains in the unit disk, arXiv:0902.3116 [math.CV], p. 29, 2009.
Tom Copeland, Flipping Functions with Permutohedra, Posted Oct 2008.
Tom Copeland, Mathemagical Forests v2, 2008.
Tom Copeland, Addendum to Mathemagical Forests, 2010.
Tom Copeland, Important formulas in combinatorics, MathOverflow answer, 2015.
Tom Copeland, Formal group laws and binomial Sheffer sequences, 2018.
Tom Copeland, Pre-Lie algebras, Cayley's analytic trees, and mathemagical forests, 2018.
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.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, p. 526 and 527.
V. Goryainov and O. Kudryavtseva, One-parameter semigroups of analytic functions, fixed points and the Koenigs function, Sbornik: Mathematics, 202:7 (2011), 971-1000.
D. Grinberg, Commutators, matrices, and an identity of Copeland, arXiv:1908.09179 [math.RA], 2019.
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.
T. Krajewski and T. Martinetti, Wilsonian renormalization, differential equations and Hopf algebras, arXiv:0806.4309 [hep-th], 2008.
Peter Luschny, Expansion A145271 (added Jul 21 2016)
MathOverflow, Important formulas in combinatorics: The combinatorics underlying iterated derivatives (infinitesimal Lie generators) for compositional inversion and flow maps for vector fields, answer by Tom Copeland to an MO question posed by Gil Kalai, 2015.
MathOverflow, Characterizing positivity of formal group laws, a MO question posed by Jair Taylor, 2018.
MathOverflow, Expansions of iterated, or nested, derivatives, or vectors--conjectured matrix computation, a MO question posed by Tom Copeland, answered by Darij Grinberg, 2019.
MathOverflow, A Leibniz-like formula for (f(x)D_x)^n f(x)?, a MO question posed by the user M.G. and answered by Tom Copeland, 2022.
MathStackExchange, Closed form for sequence A145271, posed 2014, response by Tom Copeland in 2016.
M. Mendez, Combinatorial differential operators in: Faà di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees, arXiv:1610.03602 [math.CO], p. 28 Example 6, 2016.
M. Petreolle, A. Sokal, and B. Zhu, Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes--Rogers and Thron--Rogers polynomials, with coefficientwise Hankel-total positivity, arXiv:1807.03271 [math.CO], 2020.
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 66, eqn. 9.3.
Wikipedia, Abel equation
Wikipedia, Renormaliztion Group
B. Zhu, Coefficientwise Hankel-total positivity of row-generating polynomials for the m-Jacobi-Rogers triangle, arXiv:2202.03793 [math.CO], 2022.
J. Zinn-Justin, Phase Transitions and Renormalization Group: from Theory to Numbers, Séminaire Poincaré 2, pp. 55-74, 2002.
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
A recursion relation for computing each partition polynomial of this entry from the lower order polynomials and the coefficients of the cycle index polynomials of A036039 is presented in the blog entry "Formal group laws and binomial Sheffer sequences". - Tom Copeland, Feb 06 2018
A formula for computing the polynomials of each row of this matrix is presented as T_{n,1} on p. 196 of the Ihara reference in A139605. - Tom Copeland, Mar 25 2020
Indeterminate substitutions as illustrated in A356145 lead to [E] = [L][P] = [P][E]^(-1)[P] = [P][RT] and [E]^(-1) = [P][L] = [P][E][P] = [RT][P], where [E] contains the refined Eulerian partition polynomials of this entry; [E]^(-1), A356145, the inverse set to [E]; [P], the permutahedra polynomials of A133314; [L], the classic Lagrange inversion polynomials of A134685; and [RT], the reciprocal tangent polynomials of A356144. Since [L]^2 = [P]^2 = [RT]^2 = [I], the substitutional identity, [L] = [E][P] = [P][E]^(-1) = [RT][P], [RT] = [E]^(-1)[P] = [P][L][P] = [P][E], and [P] = [L][E] = [E][RT] = [E]^(-1)[L] = [RT][E]^(-1). - Tom Copeland, Oct 05 2022
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. Similarly for (A134264, A001263, A119900), (A134685, A134991, A019538), (A133932, A111999, A007318).
Second column is A000295, subdiagonal is A000124, row sums are A000142, row lengths are A000041. - Peter Luschny, Jul 21 2016
KEYWORD
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
editing