I have a family of polynomials generated by the recurrence relation
$P_{n+1}(w) = (1+w)P_n ^{\ \prime}(w) -(3n-1 +nw)P_n(w) \\ P_1(w) =1$
The family is related to the Lambert $W$-function by its Taylor series expansion around $x=e$
$$W_0(x) = 1+\sum_{n=1}^\infty { P_n(1) \over 2^{2n-1}e^n } (x-e)^n$$
Is there any easier way to generate this polynomial family, than this recurrence relation? Or perhaps an explicit function, $P_n(w) = F(n,w)$ for $n \in \Bbb{N}$?