1,5

The lowering (or delta) operator for these polynomials is L = -1 + exp{ 2 + W[ -exp(-2) * (2+D) ] } = sum(j >= 1) A074059(j) * D^j / j! .

The raising operator is R = -x { 1 + W[ -exp(-2) * (2+D) ] } = x { 1 + sum(j >= 1) (-1)^j * PW(j-1,-2) * D^j / j! }, where PW(j-1,x) are the polynomials of A042977.

W(x) here is W_-1 in the Monir reference and, about x = 0,

W[ -exp(-2) * (2+x) ] = -[ 2 + sum(j >= 1) (-1)^j * PW(j-1,-2) * x^j / j! ] .

From the relation between delta and raising operators for

associated binomial-type polynomials, A074059 = (1,1,2,7,34,...) and S

= (1,-PW(0,-2),PW(1,-2),-PW(2,-2),...) = (1, -1, 0, -1, -2, -13, -74,

-593, -5298, ...) form a list partition transform pair (see A133314);

i.e., S and A074059 have reciprocal e.g.f.s and satisfy mutual

recursion relations. Applying Faa di Bruno's formula to L gives other

interesting integer relations between S and A074059.

F. Chapeau-Blondeau and A. Monir, Numerical Evaluation of the Lambert W Function and Application to Generation of Generalized Gaussian Noise With Exponent 1/2, IEEE Trans. on Signal Processing, Vol. 50, No. 9, Sept. 2002, p. 2160-2164.

The row polynomials P(n,t) = sum(j=1,...,n) C(n,j) * t^j satisfy exp[P(.,t) * x] = exp{ -t * [(1+x) * ln(1+x) - 2x] }, with P(0,t) = 1 and [ P(.,x) + P(.,y) ]^n = P(n,x+y) . See Mathworld and Wikipedia on Sheffer sequences and umbral calculus for other general formulae, including expansion theorems.

Cf. A134685, A135494.

Sequence in context: A134348 A074305 A151855 this_sequence A084602 A100888 A052914

Adjacent sequences: A135335 A135336 A135337 this_sequence A135339 A135340 A135341

sign,tabl

Tom Copeland (tcjpn(AT)msn.com), Feb 15 2008