  The problem of computing the class expansion of some symmetric functions
evaluated in Jucys-Murphy elements appears in different contexts, for instance
in the computation of matrix integrals. Recently, M. Lassalle gave a unified
algebraic method to obtain some induction relations on the coefficients in this
kind of expansion. In this paper, we give a simple purely combinatorial proof
of his result. Besides, using the same type of argument, we obtain new simpler
formulas. We also prove an analogous formula in the Hecke algebra of
$(S_{2n},H_n)$ and use it to solve a conjecture of S. Matsumoto on the
subleading term of orthogonal Weingarten function. Finally, we propose a
conjecture for a continuous interpolation between both problems.
