For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality. We denote by $p _ { n }$ the final value of $\left( p _ { n , N } \right) _ { N \geq 1 }$, and $P ( z ) := \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$ for all $z \in D$. Let $z \in D$. By examining the difference $\sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n } - \sum _ { n = 0 } ^ { + \infty } p _ { n , N } z ^ { n }$, prove that $$P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$$
For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality. We denote by $p _ { n }$ the final value of $\left( p _ { n , N } \right) _ { N \geq 1 }$, and $P ( z ) := \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$ for all $z \in D$.
Let $z \in D$. By examining the difference $\sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n } - \sum _ { n = 0 } ^ { + \infty } p _ { n , N } z ^ { n }$, prove that
$$P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$$