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authorGloria Mundi <gloria@gloria-mundi.eu>2024-11-16 01:24:14 +0100
committerGloria Mundi <gloria@gloria-mundi.eu>2024-11-16 01:24:14 +0100
commit98567ec798aa8ca2cfbcb85c774dd470f30e30d4 (patch)
tree5113d5cc24d1ad5f93810b6442ce584a36950dc8 /content/math/tables/numbers.tex
parentad3856a6b766087df0036de0b556f4700a6498c9 (diff)
parent8d11c6c8213f46f0fa19826917c255edd5d43cb1 (diff)
mzuenni tests
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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}