I have encountered a necessity to work with a series of the following form.

There are $N$ variables $x_1,\ldots x_N$. It is convenient to introduce monomial symmetric polynomials $m_{\lambda}$. They are defined for any power $r$ and a partition $\lambda$ of $r$ into $N$ parts, some of which can be 0, as sums of monomials with powers being elements of $\lambda$. For example, for $r=3$ and $N=3$ we have $m_{(3)}=x_1^3+x_2^3+x_3^3$, $m_{(2,1)}=x_1^2x_2+x_1^2x_3+x_2^2x_1+x_2^2x_3+x_3^2x_1+x_3^2x_2$, $m_{(1,1,1)}=x_1x_2x_3$. The power series is

$S(x_1,\ldots x_N)=\sum\limits_{r=0}^{\infty}\sum\limits_{\lambda_r}\frac{m_{\lambda_r}(x_1,\ldots x_N)}{L_{r,\lambda}}$.

Here the summation over $\lambda_r$ goes over all possible partitions of $r$ into $N$ parts with possible 0's, and $L_{r,\lambda}$ is a number of terms in $m_{\lambda_r}(x_1,\ldots x_N)$. In the example above, for $N=3$ and $r=3$ we will have, for $L_{3,(3)}=3$, $L_{3,(2,1)}=6$, $L_{3,(1,1,1)}=1$.

An alternative form of this series is

$S(x_1,\ldots x_N)=\frac{1}{N!}\sum\limits_{r=0}^{\infty}\sum\limits_{\lambda_r}N_{r,\lambda}m_{\lambda_r}(x_1,\ldots x_N)$

Here $N_{r,\lambda}$ is an order of the stabilizer of the parts of $\lambda$ under the action of the symmetric group $S_N$ acting on the variables $x_1,\ldots x_N$. Again, in the example above, for $N=3$ and $r=3$, $N_{3,(3)}=2$, $N_{3,(2,1)}=1$, $N_{3,(1,1,1)}=6$.

How can one compute a sum of such a power series? Or, maybe, a sum of some similar series with a factor of $1/L_{r,\lambda}$?