# §24.6 Explicit Formulas

The identities in this section hold for $n=1,2,\ldots$. (24.6.7), (24.6.8), (24.6.10), and (24.6.12) are valid also for $n=0$.

 24.6.1 $\mathop{B_{2n}\/}\nolimits=\sum_{k=2}^{2n+1}\frac{(-1)^{k-1}}{k}{2n+1\choose k% }\sum_{j=1}^{k-1}j^{2n},$
 24.6.2 $\mathop{B_{n}\/}\nolimits=\frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-1)^{j}j^{% n}{\binomial{n+1}{k-j}}\Bigg/{\binomial{n}{k}},$
 24.6.3 $\mathop{B_{2n}\/}\nolimits=\sum_{k=1}^{n}\frac{(k-1)!k!}{(2k+1)!}\*\sum_{j=1}^% {k}(-1)^{j-1}{2k\choose k+j}j^{2n}.$
 24.6.4 $\mathop{E_{2n}\/}\nolimits=\sum_{k=1}^{n}\frac{1}{2^{k-1}}\sum_{j=1}^{k}(-1)^{% j}{2k\choose k-j}j^{2n},$
 24.6.5 $\mathop{E_{2n}\/}\nolimits=\frac{1}{2^{n-1}}\sum_{k=0}^{n-1}(-1)^{n-k}(n-k)^{2% n}\*\sum_{j=0}^{k}{2n-2j\choose k-j}2^{j},$
 24.6.6 $\mathop{E_{2n}\/}\nolimits=\sum_{k=1}^{2n}\frac{(-1)^{k}}{2^{k-1}}{2n+1\choose k% +1}\*\sum_{j=0}^{\left\lfloor\tfrac{1}{2}k-\tfrac{1}{2}\right\rfloor}{k\choose j% }(k-2j)^{2n}.$
 24.6.7 $\mathop{B_{n}\/}\nolimits\!\left(x\right)=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0% }^{k}(-1)^{j}{k\choose j}(x+j)^{n},$
 24.6.8 $\mathop{E_{n}\/}\nolimits\!\left(x\right)=\frac{1}{2^{n}}\sum_{k=1}^{n+1}\sum_% {j=0}^{k-1}(-1)^{j}{n+1\choose k}(x+j)^{n}.$
 24.6.9 $\displaystyle\mathop{B_{n}\/}\nolimits$ $\displaystyle=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j}j^{% n},$ 24.6.10 $\displaystyle\mathop{E_{n}\/}\nolimits$ $\displaystyle=\frac{1}{2^{n}}\sum_{k=1}^{n+1}{n+1\choose k}\sum_{j=0}^{k-1}(-1% )^{j}(2j+1)^{n}.$
 24.6.11 $\mathop{B_{n}\/}\nolimits=\frac{n}{2^{n}(2^{n}-1)}\sum_{k=1}^{n}\sum_{j=0}^{k-% 1}(-1)^{j+1}{n\choose k}j^{n-1},$
 24.6.12 $\mathop{E_{2n}\/}\nolimits=\sum_{k=0}^{2n}\frac{1}{2^{k}}\sum_{j=0}^{k}(-1)^{j% }{k\choose j}(1+2j)^{2n}.$