# binomials

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##### 1: 26.3 Lattice Paths: Binomial Coefficients
###### §26.3(i) Definitions
For numerical values of $\genfrac{(}{)}{0.0pt}{}{m}{n}$ and $\genfrac{(}{)}{0.0pt}{}{m+n}{n}$ see Tables 26.3.1 and 26.3.2. …
##### 2: 24.6 Explicit Formulas
24.6.2 $B_{n}=\frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-1)^{j}j^{n}{\genfrac{(}{)}{0.% 0pt}{}{n+1}{k-j}}\Bigg{/}{\genfrac{(}{)}{0.0pt}{}{n}{k}},$
24.6.3 $B_{2n}=\sum_{k=1}^{n}\frac{(k-1)!k!}{(2k+1)!}\*\sum_{j=1}^{k}(-1)^{j-1}{2k% \choose k+j}j^{2n}.$
##### 3: 24.5 Recurrence Relations
24.5.1 $\sum_{k=0}^{n-1}{n\choose k}B_{k}\left(x\right)=nx^{n-1},$ $n=2,3,\dots$,
24.5.2 $\sum_{k=0}^{n}{n\choose k}E_{k}\left(x\right)+E_{n}\left(x\right)=2x^{n},$ $n=1,2,\dots$.
24.5.3 $\sum_{k=0}^{n-1}{n\choose k}B_{k}=0,$ $n=2,3,\dots$,
24.5.4 $\sum_{k=0}^{n}{2n\choose 2k}E_{2k}=0,$ $n=1,2,\dots$,
##### 4: 1.2 Elementary Algebra
###### §1.2(i) Binomial Coefficients
See also §26.3(i). … For complex $z$ the binomial coefficient $\genfrac{(}{)}{0.0pt}{}{z}{k}$ is defined via (1.2.6).
###### Binomial Theorem
1.2.3 $\genfrac{(}{)}{0.0pt}{}{n}{0}+\genfrac{(}{)}{0.0pt}{}{n}{1}+\dots+\genfrac{(}{% )}{0.0pt}{}{n}{n}=2^{n}.$
##### 5: 26.21 Tables
###### §26.21 Tables
Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\genfrac{(}{)}{0.0pt}{}{m}{n}$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s\left(n,k\right)$ and $S\left(n,k\right)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p\left(\mathcal{D},n\right)$ for $n$ up to 500. … It also contains a table of Gaussian polynomials up to $\genfrac{[}{]}{0.0pt}{}{12}{6}_{q}$. Goldberg et al. (1976) contains tables of binomial coefficients to $n=100$ and Stirling numbers to $n=40$.
##### 6: 24.14 Sums
24.14.3 $\sum_{k=0}^{n}{n\choose k}E_{k}\left(h\right)E_{n-k}\left(x\right)=2(E_{n+1}% \left(x+h\right)-(x+h-1)E_{n}\left(x+h\right)),$
24.14.5 $\sum_{k=0}^{n}{n\choose k}E_{k}\left(h\right)B_{n-k}\left(x\right)=2^{n}B_{n}% \left(\tfrac{1}{2}(x+h)\right),$
##### 7: 17.2 Calculus
###### §17.2(ii) Binomial Coefficients
17.2.30 $\genfrac{[}{]}{0.0pt}{}{-n}{m}_{q}=\genfrac{[}{]}{0.0pt}{}{m+n-1}{m}_{q}(-1)^{% m}q^{-mn-\genfrac{(}{)}{0.0pt}{}{m}{2}},$
###### §17.2(iii) Binomial Theorem
In the limit as $q\to 1$, (17.2.35) reduces to the standard binomial theorem … When $a=q^{m+1}$, where $m$ is a nonnegative integer, (17.2.37) reduces to the $q$-binomial series …
##### 8: 26.5 Lattice Paths: Catalan Numbers
26.5.1 $C\left(n\right)=\frac{1}{n+1}\genfrac{(}{)}{0.0pt}{}{2n}{n}=\frac{1}{2n+1}% \genfrac{(}{)}{0.0pt}{}{2n+1}{n}=\genfrac{(}{)}{0.0pt}{}{2n}{n}-\genfrac{(}{)}% {0.0pt}{}{2n}{n-1}=\genfrac{(}{)}{0.0pt}{}{2n-1}{n}-\genfrac{(}{)}{0.0pt}{}{2n% -1}{n+1}.$
26.5.5 $C\left(n+1\right)=\sum_{k=0}^{\left\lfloor n/2\right\rfloor}\genfrac{(}{)}{0.0% pt}{}{n}{2k}2^{n-2k}C\left(k\right).$
##### 9: 17.3 $q$-Elementary and $q$-Special Functions
17.3.2 $E_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}q^{\genfrac{(}{)}{0.0pt}% {}{n}{2}}x^{n}}{\left(q;q\right)_{n}}=\left(-(1-q)x;q\right)_{\infty}.$
17.3.7 $\beta_{n}\left(x,q\right)=(1-q)^{1-n}\sum_{r=0}^{n}(-1)^{r}\genfrac{(}{)}{0.0% pt}{}{n}{r}\frac{r+1}{(1-q^{r+1})}q^{rx}.$
17.3.8 $A_{m,s}\left(q\right)=q^{\genfrac{(}{)}{0.0pt}{}{s-m}{2}+\genfrac{(}{)}{0.0pt}% {}{s}{2}}\sum_{j=0}^{s}(-1)^{j}q^{\genfrac{(}{)}{0.0pt}{}{j}{2}}\genfrac{[}{]}% {0.0pt}{}{m+1}{j}_{q}\frac{(1-q^{s-j})^{m}}{(1-q)^{m}}.$