Sum of Bernoulli Numbers by Binomial Coefficients Vanishes/Examples

Example of Use of Sum of Bernoulli Numbers by Binomial Coefficients Vanishes

 * $\begin{array}{r|cccccccccc}

B_k & \dbinom n 0 & & \dbinom n 1 & & \dbinom n 2 & & \dbinom n 3 & & \dbinom n 4 & & \dbinom n 5 \\ \hline B_0 = 1             & 1 B_0  &  &  &  &  &  &  &  &  &  &  &  = 1 \\ B_1 = -\frac 1 2    & 1 B_0  & + & 2 B_1 & & &  & &  & &  &  & = 0 \\ B_2 = +\frac 1 6    & 1 B_0  & + & 3 B_1 & + & 3 B_2 &  &  & & & & &  = 0 \\ B_3 = 0             & 1 B_0  & + & 4 B_1 & + & 6 B_2 & + & 4 B_3 &  &  & & & = 0 \\ B_4 = -\frac 1 {30} & 1 B_0  & + & 5 B_1 & + & 10 B_2 & + & 10 B_3 & + & 5 B_4 &  & & = 0 \\ B_5 = 0             & 1 B_0  & + & 6 B_1 & + & 15 B_2 & + & 20 B_3 & + & 15 B_4 & + & 6 B_5 & = 0 \\ \end{array}$