%I A145271
%S A145271 1,1,1,1,1,4,1,1,11,4,7,1,1,26,34,32,15,11,1
%N 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).
%C A145271 For more detail, including connections to Legendre transformations, rooted 
               trees, A139605, A139002 and A074060, see Copeland link pg. 9.
%C A145271 For connections to the h-polynomials associated to the refined f-polynomials 
               of permutohedra see my comments in A008292 and A049019.
%H A145271 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               .
%H A145271 Tom Copeland, <a href="http://tcjpn.spaces.live.com/default.aspx">Flipping 
               Functions with Permutohedra</a> Posted Oct. 2008 [From Tom Copeland 
               (tcjpn(AT)msn.com), Oct 08 2008]
%H A145271 Tom Copeland, <a href="http://tcjpn.spaces.live.com/default.aspx">Mathemagical 
               Forests v2</a> Posted June 2008
%e A145271 Let R = g(x)d/dx then
%e A145271 R^0 g(x) = 1 (0')^1
%e A145271 R^1 g(x) = 1 (0')^1 (1')^1
%e A145271 R^2 g(x) = 1 (0')^1 (1')^2 + 1 (0')^2 (2')^1
%e A145271 R^3 g(x) = 1 (0')^1 (1')^3 + 4 (0')^2 (1')^1 (2')^1 + 1 (0')^3 (3')^1
%e A145271 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
%e A145271 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.
%e A145271 The partitions are in reverse order to those in Abramowitz and Stegun 
               pg. 831. Summing over coefficients with like powers of (0') gives 
               A008292.
%Y A145271 Sequence in context: A140711 A164366 A121692 this_sequence A147564 A090981 
               A087903
%Y A145271 Adjacent sequences: A145268 A145269 A145270 this_sequence A145272 A145273 
               A145274
%K A145271 easy,nonn,tabf
%O A145271 0,6
%A A145271 Tom Copeland (tcjpn(AT)msn.com), Oct 06 2008, Oct 08 2008