Define a variant of Stirling numbers (of the second kind) by $ S(n,k)=S(n-1,k-1)+k^2 S(n-1,k) $ with $ S(n,0)=[n=0] $ and $ S(0,k)=[k=0] $ or equivalently by the generating function $\sum {S(n,k)x^n } = \frac{{x^k }}{{\prod\limits_{j = 1}^k {(1 - j^2 x)} }}.$ Have these numbers been studied in the literature?
There are various connections with the Genocchi numbers $$(G_{2n} )_{n \ge 1} = (1,1,3,17,155,2073, \cdots ), $$
defined by the generating function $z\frac{{1 - e^z }}{{1 + e^z }} = \sum {( - 1)^n G_{2n} } \frac{{z^{2n} }}{{(2n)!}}.$ E.g. the identity $\sum\limits_{k = 0}^n {( - 1)^k } k!(k + 1)!S(n,k+1) = ( - 1)^{n - 1} G_{2n} .$ Are such formulas known?
Edit: I withdraw this question, since I have observed that it has already been answered in Problem 5.8 of Stanley, Enumerative combinatorics 2.
