Sum of Bernoulli Numbers by Binomial Coefficients Vanishes/Examples

Example of Recurrence Relation on Bernoulli Numbers
For $n \in \N_{>0}$:
 * $\displaystyle \sum_{k \mathop = 0}^n \binom {n + 1} k B_k = 0$

Proof

 * $\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}$

Also see

 * Sum of Bernoulli Numbers by Binomial Coefficients Vanishes