  We consider a deformation of Kerov character polynomials, linked to Jack
symmetric functions. It has been introduced recently by M. Lassalle, who
formulated several conjectures on these objects, suggesting some underlying
combinatorics. We give a partial result in this direction, showing that some
quantities are polynomials in the Jack parameter $\alpha$ with prescribed
degree. Our result has several interesting consequences in various directions.
Firstly, we give a new proof of the fact that the coefficients of Jack
polynomials expanded in the monomial or power-sum basis depend polynomially in
$\alpha$. Secondly, we describe asymptotically the shape of random Young
diagrams under some deformation of Plancherel measure.
