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A139605
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Weights for expansion of (f(x)D_x)^n : coefficients of A-polynomials of Comtet.
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8
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1, 1, 1, 1, 1, 3, 1, 1, 4, 1, 7, 4, 6, 1, 1, 11, 4, 7, 1, 15, 30, 5, 25, 10, 10, 1, 1, 26, 34, 32, 11, 15, 1, 31, 146, 57, 34, 6, 90, 120, 15, 65, 20, 15, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,6
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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
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REFERENCES
| F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-like Structures, (1997), Cambridge University Press, p. 386
L. Comtet, Une formule explicite pour puisances successives de l'operateur de derivation de Lie, Comtes Rendus Acad. Sc. Paris, Serie A tome 276 (1973), pp. 165 - 168.
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) 1-133. eq. (I.2.18).
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LINKS
| T. Copeland, Mathemagical Forests (June 2008)
Kentaro Ihara, Derivations and automorphisms on non-commutative power series, Journal of Pure and Applied Algebra, Volume 216, Issue 1, January 2012, Pages 192-201; doi:10.1016/j.jpaa.2011.06.004
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CROSSREFS
| Sequence in context: A132442 A074927 A191780 * A098712 A023579 A023577
Adjacent sequences: A139602 A139603 A139604 * A139606 A139607 A139608
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KEYWORD
| nonn
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AUTHOR
| Tom Copeland (tcjpn(AT)msn.com), Jun 12 2008
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