Sum of Bernoulli Numbers by Binomial Coefficients Vanishes/Examples


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

B_N & \dbinom {n+1} {0} & & \dbinom {n+1} {1} & & \dbinom {n+1} {2} & & \dbinom {n+1} {3} & & \dbinom {n+1} {4} & & \dbinom {n+1} {5} & & \cdots \\ \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\\

\cdots                 & \cdots & + & \cdots & + & \cdots & + & \cdots & + & \cdots & + & \cdots & + & \cdots & = 0 \\

\end{array}$