  Each numerical sequence $\left( {{b}_{0}},{{b}_{1}},{{b}_{2}},... \right)$
with the generating function $B\left( x \right)$ defines the pseudo-involution
in the Riordan group $\left( 1,xg\left( x \right) \right)$ such that $g\left( x
\right)=1+xg\left( x \right)B\left( {{x}^{2}}g\left( x \right) \right)$. In the
present paper we realize a simple idea: express the coefficients of the series
${{g}^{m}}\left( x \right)$ in terms of the coefficients of the series $B\left(
x \right)$. Obtained expansion has a bright combinatorial character, sheds
light on the connection of the pseudo-involution in the Riordan group with the
generalized binomial series, and is also useful for finding the series $g\left(
x \right)$ by the given series $B\left( x \right)$. We compare this expansion
with the similar expansion for the sequence $\left( 1,{{a}_{1}},{{a}_{2}},...
\right)$ with the generating function $A\left( x \right)$ such that $g\left( x
\right)=A\left( xg\left( x \right) \right)$.
