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authormzuenni <michi.zuendorf@gmail.com>2024-09-08 22:16:36 +0200
committermzuenni <michi.zuendorf@gmail.com>2024-09-08 22:16:36 +0200
commit45d95c45013bf4ff73570c94c58b7f0212ccdf26 (patch)
treee47f9b16f44acef5f2517655fe2a16f8108f5af1 /content/math/tables/numbers.tex
parentdf963645ca0c5d0bed4fb9c02e93233dcfd53dae (diff)
moved stuff
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-\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}