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Diffstat (limited to 'content/math/tables/binom.tex')
| -rw-r--r-- | content/math/tables/binom.tex | 31 |
1 files changed, 15 insertions, 16 deletions
diff --git a/content/math/tables/binom.tex b/content/math/tables/binom.tex index 878a6b0..9fc9ae3 100644 --- a/content/math/tables/binom.tex +++ b/content/math/tables/binom.tex @@ -1,28 +1,27 @@ -\begin{tabularx}{\linewidth}{|XXXX|} +\begin{expandtable} +\begin{tabularx}{\linewidth}{|C|} \hline - \multicolumn{4}{|c|}{Binomialkoeffizienten} \\ - \hline - \multicolumn{4}{|c|}{ $\frac{n!}{k!(n - k)!} \hfill=\hfill \binom{n}{k} \hfill=\hfill \binom{n}{n - k} \hfill=\hfill \frac{n}{k}\binom{n - 1}{k - 1} \hfill=\hfill \frac{n-k+1}{k}\binom{n}{k - 1} \hfill=\hfill - \binom{n - 1}{k} + \binom{n - 1}{k - 1} \hfill=\hfill + \frac{k+1}{n-k}\binom{n}{k + 1} \hfill=\hfill$\\ + + $\binom{n - 1}{k - 1} + \binom{n - 1}{k} \hfill=\hfill + \binom{n + 1}{k + 1} - \binom{n}{k + 1} \hfill=\hfill (-1)^k \binom{k - n - 1}{k} \hfill\approx\hfill - 2^{n} \cdot \frac{2}{\sqrt{2\pi n}}\cdot\exp\left(-\frac{2(x - \frac{n}{2})^2}{n}\right)$ - } \\ + 2^{n} \cdot \frac{2}{\sqrt{2\pi n}}\cdot\exp\left(-\frac{2(x - \frac{n}{2})^2}{n}\right)$\\ \grayhline - $\sum\limits_{k = 0}^n \binom{n}{k} = 2^n$ & - $\sum\limits_{k = 0}^n \binom{k}{m} = \binom{n + 1}{m + 1}$ & - $\sum\limits_{i = 0}^n \binom{n}{i}^2 = \binom{2n}{n}$ & - $\sum\limits_{k = 0}^n\binom{r + k}{k} = \binom{r + n + 1}{n}$\\ + $\sum\limits_{k = 0}^n \binom{n}{k} = 2^n\hfill + \sum\limits_{k = 0}^n \binom{k}{m} = \binom{n + 1}{m + 1}\hfill + \sum\limits_{i = 0}^n \binom{n}{i}^2 = \binom{2n}{n}\hfill + \sum\limits_{k = 0}^n\binom{r + k}{k} = \binom{r + n + 1}{n}$\\ - $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n - k}{m - k}$ & - $\sum\limits_{k = 0}^n \binom{r}{k}\binom{s}{n - k} = \binom{r + s}{n}$ & - \multicolumn{2}{l|}{ - $\sum\limits_{i = 1}^n \binom{n}{i} F_i = F_{2n} \quad F_n = n\text{-th Fib.}$ - }\\ + $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n - k}{m - k}\hfill + \sum\limits_{k = 0}^n \binom{r}{k}\binom{s}{n - k} = \binom{r + s}{n}\hfill + \sum\limits_{i = 1}^n \binom{n}{i} \mathit{Fib}_i = \mathit{Fib}_{2n}$\\ \hline \end{tabularx} +\end{expandtable} |