I am stack on this question. ( I wanted to prove that function, representation of which is exactly this series, has one extremal point).
Let $A_j=\{n \in N: n_1+\ldots n_k=n \quad \mbox{and } \quad n_2+2n_3+\ldots +(k+1)n_k=j\}$
Help me please to show if the following series is alternating with decreasing terms? $$ S=c_0+\frac{c_1}{n}+\sum_{i=2}^{\infty}\frac{c_i}{n^i}, $$ where $$ c_{j}=\sum_{i=0}^j\frac{b_i}{6^{j-i}(j-i)!} \quad \mbox{with} \quad b_j=n!\sum_{n \in A_j}\left(\frac{-1}{n}\right)^j\frac{1}{\prod_{i=1}^k((2i-1)!)^{n_i}n_i!} $$
