  A new recursion procedure for deriving renormalized perturbation expansions
for the one-dimensional anharmonic oscillator is offered. Based upon the
$\hbar$-expansions and suitable quantization conditions, the recursion formulae
obtained have the same simple form both for ground and excited states and can
be easily applied to any renormalization scheme. As an example, the
renormalized expansions for the sextic anharmonic oscillator are considered.
