

A139605


Weights for expansion of iterated derivatives, powers of a Lie derivative, or infinitesimal generator in vector form, (f(x)D_x)^n: coefficients of Apolynomials of Comtet.


12



1, 1, 1, 1, 1, 3, 1, 1, 4, 1, 4, 7, 6, 1, 1, 7, 4, 11, 1, 5, 30, 15, 10, 25, 10, 1, 1, 11, 15, 32, 34, 26, 1, 6, 57, 34, 146, 31, 15, 120, 90, 20, 65, 15, 1, 1, 16, 26, 15, 76, 192, 34, 122, 180, 57, 1, 7, 98, 140, 406, 462, 588, 63, 21, 252, 154, 896, 301
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OFFSET

1,6


COMMENTS

This entry and the references differ slightly among themselves in the order of coefficients for higher order terms. Table on p. 167 of Comtet has at least one index error.
Let F[FI(x)]=FI[F(x)]=x (i.e., F and FI are a comp. inverse pair) about x=0 with F(0)=FI(0)=0. Define f(x)=1/[dFI(x)/dx], then for w(x) analytic about x=0, exp[t f(x)d/dx] w(x)= w{F[t+FI(x)]}=q(t,x)
with q{t,F[s+FI(x)]}=q(t+s,x). See A145271 for w(x)= x and note that A145271 is embedded in A139605. E.g.f. of the binomial Sheffer sequence associated to F(x) is exp[x f(z)d/dz] exp(t*z)= exp{t*F[x+FI(z)]} evaluated at z=0.  Tom Copeland, Oct 19 2011
dq(t,x)/dt  f(x)dq(t,x)/dx = 0, so (1,f(x)) gives the components of a vector orthogonal to the gradient of q and therefore tangent to the contour of q at (t,x).  Tom Copeland, Oct 26 2011
The formula exp[t f(x)d/dx] w(x)= w{F[t+FI(x)]} above is implicit in the chain rule formulas on pages 10 and 12 of Mathemagical Forests. Another derivation is alluded to in the Dattoli reference in A080635 (repeated below).  Tom Copeland, Nov 28 2011
Let f(x) and g(x) be two infinitely differential functions. Denote f_0 = f(x), f_1 = df_0/dx, f_2 = df_1/dx, and so on. Same with g_0 = g(x). Define the linear operator L(u(x)) = g(x) * du(x)/dx. Denote F_1 = L(f(x)), F_2 = L(F_1), and so on. When n>0, F_n is a linear combination of f_1, ..., f_n where each f_k is multiplied by a homogeneous polynomial (P(n,k)) of degree n in g_0, ..., g_{n1}. The triangle of the sum of the coefficients of P(n,k) is A130534.  Michael Somos, Mar 23 2014
Triangle with row n length A000070(n+1) and row n consists of the coefficients: P(n,1), ..., P(n,n). The order of coefficients in P(n,k) is Abramowitz and Stegun order for partitions of nk with parts g_1, ..., g_{nk}.  Michael Somos, Mar 23 2014
A130534(n,k) gives the number of rooted trees with (k+1) trunks that are associated with D^(k+1) in the forest of "naturally grown" rooted trees with (n+2) nodes, or vertices, that are associated with R^(n+1) in the example below. Cf. MF link.  Tom Copeland, Mar 23 2014


REFERENCES

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Treelike Structures, (1997), Cambridge University Press, p. 386
L. Comtet, Une formule explicite pour les puissances successives de l'opĂ©rateur de dĂ©rivation de Lie, Comptes Rendus Acad. Sc. Paris, Serie A tome 276 (1973), pp. 165  168.
H. Davis, The Theory of Linear Operators, (1936), The Principia Press, p. 13


LINKS

Table of n, a(n) for n=1..68.
Tom Copeland, Mathemagical Forests v2, 2008.
T. Copeland, Infinigens, the Pascal Triangle, and the Witt and Virasoro Algebras, 2012.
Tom Copeland, PreLie algebras, Cayley's analytic trees, and mathemagical forests, 2018.
G. Datolli, P. L. Ottaviani, A. Torre and L. Vazquez, Evolution operator equations: integration with algebraic and finite differences methods.[...], La Rivista del Nuovo Cimento 20,2 (1997) 1133. eq. (I.2.18).
D. Grinberg, Commutators, matrices, and an identity of Copeland, arXiv:1908.09179 [math.RA], 2019.
Kentaro Ihara, Derivations and automorphisms on noncommutative power series, Journal of Pure and Applied Algebra, Volume 216, Issue 1, January 2012, Pages 192201.
T. Mansour and M. Shork, The generalized Touchard polynomials revisited, Journal of Applied Mathematics and Computation, Volume 219, Issue 19, June 2013, Pages 99789991.
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, Expansions of iterated, or nested, derivatives, or vectorsconjectured matrix computation, a MO question posed by Tom Copeland, answered by Darij Grinberg, 2019.


FORMULA

Equivalent matrix computation: Multiply the nth diagonal (with n=0 the main diagonal) of the lower triangular Pascal matrix by f_n = (d/dx)^n f(x) to obtain the matrix VP with VP(n,k) = binomial(n,k) f_(nk). Then R^n = (1, 0, 0, 0,..) [VP * S]^n (1, D, D^2, ..)^T, where S is the shift matrix A129185, representing differentiation in the basis x^n/n!. Cf. A145271.  Tom Copeland, Jul 17 2016
A formula for the coefficients of this matrix is presented in Ihara, obtained from Comtet.  Tom Copeland, Mar 25 2020
Elaborating on my 2011 comments: Let exp[x F(t)] = exp[t p.(x)] be the e.g.f. for the binomial Sheffer sequence of polynomials (p.(x))^n = p_n(x). Then, evaluated at x = t = 0, the coefficient p_(n,k) = (D_x^k/k!) p_n(x) = D_t^n [F(t)]^k/k! = (f(x)D_x)^n x^k/k! = R^n x^k/k!, and so p_(n,k) is the coefficient of D^k of the operator R^n below evaluated at x=0.  Tom Copeland, May 14 2020


EXAMPLE

Let R = f(x)d/dx = f(x)D and (j,k) = [(d/dx)^j f(x)]^k ; then
R^0 = 1.
R^1 = (0,1)D.
R^2 = (0,1)(1,1)D + (0,2)D^2.
R^3 = [(0,1)(1,2) + (0,2)(2,1)]D + 3 (0,2)(1,1)D^2 + (0,3)D^3.
R^4 = [(0,1)(1,3) + 4 (0,2)(1,1)(2,1) + (0,3)(3,1)]D +
[7 (0,2)(1,2) + 4 (0,3)(2,1)]D^2 + 6 (0,3)(1,1)D^3 + (0,4)D^4.  Tom Copeland, Jun 12 2008
R^5 = [(0,1)(1,4) + 11 (0,2)(1,2)(2,2) + 4 (0,3)(2,2) + (0,4)(4,1) + 7 (0,3)(1,1)(3,1)]D + [15 (0,2)(1,3) + 30 (0,3)(1,1)(2,1) + 5 (0,4)(1,3)]D^2 + [25 (0,3)(1,2) + 10 (0,4)(2,1) + 25 (0,3)(1,2)]D^3 + 10 (0,4)(1,1)D^4 + (0,5)D^5.  Tom Copeland, Jul 17 2016
R^6 = [(0,1)(1,5) + 26 (0,2)(1,3)(2,1) + 34 (0,3)(1,1)(2,2) + 32 (0,3)(1,2)(3,1) + 11 (0,4)(1,1)(4,1) + 15 (0,4)(2,1)(3,1) + (0,5)(1,5)]D + [31 (0,2)(1,4) + 146 (0,3)(1,2)(2,1) + 57 (0,4)(1,1)(3,1) + 34 (0,4)(2,2) + 6 (0,5)(4,1)]D^2 + [90 (0,3)(1,3) + 120 (0,4)(1,1)(2,1) + 15 (0,5)(3,1)]D^3 + [65 (0,4)(1,2) + 20 (0,5)(2,1)]D^4 + 15 (0,5)(1,1)D^5 + (0,6)D^6.  Tom Copeland, Jul 17 2016

F_1 = (1*g_0) * f_1, F_2 = (1*g_0*g_1) * f_1 + (1*g_0^2) * f_2, F_3 = (1*g_0*g_1^2 + 1*g_0^2*g_2) * f_1 + (3*g_0^2*g_1) * f_2 + (1*g_0^3) * f_3.  Michael Somos, Mar 23 2014
P(4,2) = 4*g0^3*g2 + 7*g0^2*g1^2. P(5,2) = 5*g0^4*g3 + 30*g0^3*g1*g2 + 15*g0^2*g1^3.  Michael Somos, Mar 23 2014
1
1 , 1
1 1 , 3 , 1
1 4 1 , 4 7 , 6 , 1
1 7 4 11 1, 5 30 15 , 10 25 , 10 , 1
1 11 15 32 34 26 1 , 6 57 34 146 31 , 15 120 90 , 20 65 , 15 , 1


CROSSREFS

Cf. A000070 (number of distinct terms for each order).
Cf. A130534 (sum of numerical coefficients of the derivatives).
Cf. A129185, A145271.
Sequence in context: A138684 A132442 A074927 * A191780 A098712 A264490
Adjacent sequences: A139602 A139603 A139604 * A139606 A139607 A139608


KEYWORD

nonn,tabf


AUTHOR

Tom Copeland, Jun 12 2008


EXTENSIONS

Title expanded by Tom Copeland, Mar 17 2014
Sequence terms rearranged in Abramowitz and Stegun order by Michael Somos, Mar 23 2014


STATUS

approved



