Find the sum $$2\sum_{k=1}^{\infty}\sum_{i=0}^{2k-1}\dfrac{\binom{2k}{i}\cdot B_{i}\cdot(m-1)^{2k-i}}{2k(2k-1)m^{2k-1}}$$
where $B_{i}$ is Bernoulli numbers.
my idea: since $$2\sum_{k=1}^{\infty}\sum_{i=0}^{2k-1}\dfrac{\binom{2k}{i}\cdot B_{i}\cdot(m-1)^{2k-i}}{2k(2k-1)m^{2k-1}}=2\sum_{k=1}^{\infty}\dfrac{1}{(2k-1)m^{2k-1}}\sum_{i=1}^{m-1}i^{2k-1}$$
It is said this problem is from Normal distribution,deriration by de Moivre 1721-1733.But I can't find it.Thank you