For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ 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 $(p_{n,N})_{N \geq 1}$. We fix $\ell \in \mathbf{N}$ and $x \in [0,1[$. Using the result of the previous question, establish the bound $\sum_{n=0}^{\ell} p_n x^n \leq P(x)$. Deduce the radius of convergence of the power series $\sum_n p_n z^n$.
For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ 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 $(p_{n,N})_{N \geq 1}$.
We fix $\ell \in \mathbf{N}$ and $x \in [0,1[$. Using the result of the previous question, establish the bound $\sum_{n=0}^{\ell} p_n x^n \leq P(x)$. Deduce the radius of convergence of the power series $\sum_n p_n z^n$.