# §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 $B_{2n}=\sum_{k=2}^{2n+1}\frac{(-1)^{k-1}}{k}{2n+1\choose k}\sum_{j=1}^{k-1}j^{% 2n},$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E1 Encodings: TeX, pMML, png See also: Annotations for 24.6 and 24
 24.6.2 $B_{n}=\frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-1)^{j}j^{n}{\genfrac{(}{)}{0.% 0pt}{}{n+1}{k-j}}\Bigg{/}{\genfrac{(}{)}{0.0pt}{}{n}{k}},$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E2 Encodings: TeX, pMML, png See also: Annotations for 24.6 and 24
 24.6.3 $B_{2n}=\sum_{k=1}^{n}\frac{(k-1)!k!}{(2k+1)!}\*\sum_{j=1}^{k}(-1)^{j-1}{2k% \choose k+j}j^{2n}.$
 24.6.4 $E_{2n}=\sum_{k=1}^{n}\frac{1}{2^{k-1}}\sum_{j=1}^{k}(-1)^{j}{2k\choose k-j}j^{% 2n},$ ⓘ Symbols: $E_{\NVar{n}}$: Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E4 Encodings: TeX, pMML, png See also: Annotations for 24.6 and 24
 24.6.5 $E_{2n}=\frac{1}{2^{n-1}}\sum_{k=0}^{n-1}(-1)^{n-k}(n-k)^{2n}\*\sum_{j=0}^{k}{2% n-2j\choose k-j}2^{j},$ ⓘ Symbols: $E_{\NVar{n}}$: Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E5 Encodings: TeX, pMML, png See also: Annotations for 24.6 and 24
 24.6.6 $E_{2n}=\sum_{k=1}^{2n}\frac{(-1)^{k}}{2^{k-1}}{2n+1\choose k+1}\*\sum_{j=0}^{% \left\lfloor\tfrac{1}{2}k-\tfrac{1}{2}\right\rfloor}{k\choose j}(k-2j)^{2n}.$
 24.6.7 $B_{n}\left(x\right)=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j% }(x+j)^{n},$
 24.6.8 $E_{n}\left(x\right)=\frac{1}{2^{n}}\sum_{k=1}^{n+1}\sum_{j=0}^{k-1}(-1)^{j}{n+% 1\choose k}(x+j)^{n}.$
 24.6.9 $\displaystyle B_{n}$ $\displaystyle=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j}j^{% n},$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E9 Encodings: TeX, pMML, png See also: Annotations for 24.6 and 24 24.6.10 $\displaystyle E_{n}$ $\displaystyle=\frac{1}{2^{n}}\sum_{k=1}^{n+1}{n+1\choose k}\sum_{j=0}^{k-1}(-1% )^{j}(2j+1)^{n}.$ ⓘ Symbols: $E_{\NVar{n}}$: Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6, §24.6 Permalink: http://dlmf.nist.gov/24.6.E10 Encodings: TeX, pMML, png See also: Annotations for 24.6 and 24
 24.6.11 $B_{n}=\frac{n}{2^{n}(2^{n}-1)}\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j+1}{n% \choose k}j^{n-1},$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E11 Encodings: TeX, pMML, png See also: Annotations for 24.6 and 24
 24.6.12 $E_{2n}=\sum_{k=0}^{2n}\frac{1}{2^{k}}\sum_{j=0}^{k}(-1)^{j}{k\choose j}(1+2j)^% {2n}.$ ⓘ Symbols: $E_{\NVar{n}}$: Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6, §24.6 Permalink: http://dlmf.nist.gov/24.6.E12 Encodings: TeX, pMML, png See also: Annotations for 24.6 and 24