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  &  &  &  &  &  & \cdots & = 1\\

B_1 = - \frac {1} {2} & 1 B_0  & +2 B_1 &  &  &  &  & \cdots & = 0\\

B_2 = + \frac {1} {6} & 1 B_0  & +3 B_1 & + 3 B_2 &  &  &  & \cdots & = 0\\

B_3 = 0 & 1 B_0  & +4 B_1 & +6 B_2 & + 4 B_3 &  &  & \cdots & = 0\\

B_4 = - \frac {1} {30} & 1 B_0  & +5 B_1 & +10 B_2 & +10 B_3 & +5 B_4 &  & \cdots & = 0\\

B_5 = 0 & 1 B_0  & +6 B_1 & +15 B_2 & +20 B_3 & +15 B_4 & +6 B_5 & \cdots & = 0\\

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

\end{array}$