diff options
Diffstat (limited to 'math/tables/numbers.tex')
| -rw-r--r-- | math/tables/numbers.tex | 59 |
1 files changed, 59 insertions, 0 deletions
diff --git a/math/tables/numbers.tex b/math/tables/numbers.tex new file mode 100644 index 0000000..1dc9f38 --- /dev/null +++ b/math/tables/numbers.tex @@ -0,0 +1,59 @@ +\begin{expandtable} +\begin{tabularx}{\linewidth}{|l|X|} + \hline + \multicolumn{2}{|c|}{Berühmte Zahlen} \\ + \hline + \textsc{Fibonacci} & + $f(0) = 0 \quad + f(1) = 1 \quad + f(n+2) = f(n+1) + f(n)$ \\ + \grayhline + + \textsc{Catalan} & + $C_0 = 1 \qquad + C_n = \sum\limits_{k = 0}^{n - 1} C_kC_{n - 1 - k} = + \frac{1}{n + 1}\binom{2n}{n} = \frac{2(2n - 1)}{n+1} \cdot C_{n-1}$ \\ + \grayhline + + \textsc{Euler} I & + $\eulerI{n}{0} = \eulerI{n}{n-1} = 1 \qquad + \eulerI{n}{k} = (k+1) \eulerI{n-1}{k} + (n-k) \eulerI{n-1}{k-1} $ \\ + \grayhline + + \textsc{Euler} II & + $\eulerII{n}{0} = 1 \quad + \eulerII{n}{n} = 0 \quad$\\ + & $\eulerII{n}{k} = (k+1) \eulerII{n-1}{k} + (2n-k-1) \eulerII{n-1}{k-1}$ \\ + \grayhline + + \textsc{Stirling} I & + $\stirlingI{0}{0} = 1 \qquad + \stirlingI{n}{0} = \stirlingI{0}{n} = 0 \qquad + \stirlingI{n}{k} = \stirlingI{n-1}{k-1} + (n-1) \stirlingI{n-1}{k}$ \\ + \grayhline + + \textsc{Stirling} II & + $\stirlingII{n}{1} = \stirlingII{n}{n} = 1 \qquad + \stirlingII{n}{k} = k \stirlingII{n-1}{k} + \stirlingII{n-1}{k-1} = + \frac{1}{k!} \sum\limits_{j=0}^{k} (-1)^{k-j}\binom{k}{j}j^n$\\ + \grayhline + + \textsc{Bell} & + $B_1 = 1 \qquad + B_n = \sum\limits_{k = 0}^{n - 1} B_k\binom{n-1}{k} + = \sum\limits_{k = 0}^{n}\stirlingII{n}{k}$\\ + \grayhline + + \textsc{Partitions} & + $p(0,0) = 1 \quad + p(n,k) = 0 \text{ für } k > n \text{ oder } n \leq 0 \text{ oder } k \leq 0$ \\ + & $p(n,k) = p(n-k,k) + p(n-1,k-1)$\\ + \grayhline + + \textsc{Partitions} & + $f(0) = 1 \quad f(n) = 0~(n < 0)$ \\ + & $f(n)=\sum\limits_{k=1}^\infty(-1)^{k-1}f(n - \frac{k(3k+1)}{2})+\sum\limits_{k=1}^\infty(-1)^{k-1}f(n - \frac{k(3k-1)}{2})$\\ + + \hline +\end{tabularx} +\end{expandtable} |
