0,6

For more detail, including connections to Legendre transformations, rooted trees, A139605, A139002 and A074060, see Copeland link pg. 9.

For connections to the h-polynomials associated to the refined f-polynomials of permutohedra see my comments in A008292 and A049019.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 .

Tom Copeland, Flipping Functions with Permutohedra Posted Oct. 2008 [From Tom Copeland (tcjpn(AT)msn.com), Oct 08 2008]

Tom Copeland, Mathemagical Forests v2 Posted June 2008

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

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 pg. 831. Summing over coefficients with like powers of (0') gives A008292.

Sequence in context: A140711 A164366 A121692 this_sequence A147564 A090981 A087903

Adjacent sequences: A145268 A145269 A145270 this_sequence A145272 A145273 A145274

easy,nonn,tabf

Tom Copeland (tcjpn(AT)msn.com), Oct 06 2008, Oct 08 2008