

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.


11



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.
T. Copeland, Mathemagical Forests (June 2008)
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).
Kentaro Ihara, Derivations and automorphisms on noncommutative power series, Journal of Pure and Applied Algebra, Volume 216, Issue 1, January 2012, Pages 192201.


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

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).
Sequence in context: A138684 A132442 A074927 * A191780 A098712 A023579
Adjacent sequences: A139602 A139603 A139604 * A139606 A139607 A139608


KEYWORD

nonn


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



