kozyさんの級数

$$\begin{array}{rl}\sum _{1\le a_1\le \cdots \le a_k}\frac{1}{a_1\cdots a_k{2}^{a_k}}& =\sum _{1\le a_1\le \cdots \le a_k}\frac{1}{a_2\cdots a_k{2}^{a_k}}{\int }_0^1{t}^{a_1-1}dt\\ & ={\int }_0^1\sum _{1\le a_2\le \cdots \le a_k}\frac{1}{a_2\cdots a_k{2}^{a_k}}\sum _{a_1=1}^{a_2}{t}^{a_1-1}dt\\ & ={\int }_0^1\sum _{1\le a_2\le \cdots \le a_k}\frac{1}{a_2\cdots a_k{2}^{a_k}}\frac{1-t^{a_2}}{1-t}dt\\ & ={\int }_0^1\frac{dt}{1-t}\sum _{1\le a_2\le \cdots \le a_k}\frac{1-t^{a_2}}{a_2\cdots a_k{2}^{a_k}}\\ & ={\int }_0^1\frac{dt}{1-t}\sum _{1\le a_2\le \cdots \le a_k}\frac{1}{a_3\cdots a_k{2}^{a_k}}{\int }_t^1{u}^{a_2-1}du\\ & ={\int }_0^1\frac{dt}{1-t_1}{\int }_{t_1}^1\frac{d{t}_2}{1-t_2}\cdots {\int }_{t_{k-2}}^1\frac{d{t}_{k-1}}{1-t_{k-1}}\sum _{1\le a_k}\frac{1-t_{k-1}^{a_k}}{a_k{2}^{a_k}}\\ & ={\int }_0^1\frac{dt}{1-t_1}\cdots {\int }_{t_{k-2}}^1\frac{d{t}_{k-1}}{1-t_{k-1}}{\int }_{t_{k-1}}^1\sum _{1\le a_k}\frac{t_k^{a_k-1}}{2^{a_k}}d{t}_k\\ & ={\int }_0^1\frac{dt}{1-t_1}\cdots {\int }_{t_{k-2}}^1\frac{d{t}_{k-1}}{1-t_{k-1}}{\int }_{t_{k-1}}^1\frac{d{t}_k}{2-t_k}\\ & ={\int }_{0<t_1<\cdots <t_k<1}\frac{d{t}_1}{1-t_1}\cdots \frac{d{t}_{k-1}}{1-t_{k-1}}\frac{d{t}_k}{2-t_k}\\ & ={\int }_{0<u_k<\cdots <u_1<1}\frac{d{u}_1}{u_1}\cdots \frac{d{u}_{k-1}}{u_{k-1}}\frac{d{u}_k}{1+u_k}(t_i\mapsto 1-u_i)\\ & =-{\int }_{0<u_k<\cdots <u_1<1}\frac{d{u}_1}{u_1}\cdots \frac{d{u}_{k-1}}{u_{k-1}}\frac{d{u}_k}{-1-u_k}\\ & =-\zeta (\bar{k})\\ & =\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n^k}\end{array}$$