Sum of Bernoulli Numbers by Binomial Coefficients Vanishes/Examples

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

$\forall n \in \Z_{>1}: \displaystyle \sum_{k \mathop = 0}^{n - 1} \binom n k B_k = 0$

where $B_k$ denotes the $k$th Bernoulli number.


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