1,3

This is a reordered version of A008299 read along the diagonals (see table on p. 222 in Comtet), and a row-reversed version of a table on p. 92 in the Ward reference. A134685 is a refinement of the Ward table. The first and second columns are A001147 and A000457, and appear in the diagonals of several OEIS entries. The polynomials also appear in Carlitz (p. 85), Drake et. al (p. 8), and Smiley (p. 7).

First few polynomials are

P(0,t) = 0

P(1,t) = 1

P(2,t) = t

P(3,t) = t + 3 t^2

P(4,t) = t + 10 t^2 + 15 t^3

P(5,t) = t + 25 t^2 + 105 t^3 + 105 t^4

L. Carlitz, The coefficients in an asymptotic expansion and certain related numbers, Duke Math. J., vol 35 (1968), p. 83-90.

L. Comtet, Advanced Combinatorics, Reidel, 1974.

B. Drake, I. M. Gessel and G. Xin, Three proofs and a generalization of the Goulden-Litsyn-Shevelev conjecture ..., J. Integer Sequences, Vol. 10 (2007), #07.3.7.

M. Ward, The representations of Stirling's numbers and Stirling's polynomials as sums of factorials, Amer. J. Math., 56 (1934), p. 87-95.

L. M. Smiley, Completion of a Rational Function Sequence of Carlitz

E.g.f. for the polynomials is A(x,t) = (x-t)/(t+1) + T{ (t/(t+1)) * exp[(x-t)/(t+1)] }, where T(x) is the Tree function, the e.g.f. of A000169. The compositional inverse in x (about x = 0) is B(x) = x + -t * [exp(x) - x - 1]. Special case t = 1 gives e.g.f. for A000311. These results are a special case of A134685 with u(x) = B(x).

Sequence in context: A084178 A068438 A064060 this_sequence A095327 A048953 A119632

Adjacent sequences: A134988 A134989 A134990 this_sequence A134992 A134993 A134994

nonn,tabl

Tom Copeland (tcjpn(AT)msn.com), Feb 05 2008