From 5ab8a5088b729a9953b8dff1b2a985dc8fb2098b Mon Sep 17 00:00:00 2001
From: mzuenni <michi.zuendorf@gmail.com>
Date: Mon, 27 Jun 2022 17:19:28 +0200
Subject: updated tcr

---
 math/tables/numbers.tex | 59 +++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 59 insertions(+)
 create mode 100644 math/tables/numbers.tex

(limited to 'math/tables/numbers.tex')

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}
-- 
cgit v1.2.3

