moved stuff

8 files changed, 81 insertions, 143 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}
diff --git a/content/math/tables/composite.tex b/content/math/tables/composite.tex
index c261db1..7a6ab09 100644
--- a/content/math/tables/composite.tex
+++ b/content/math/tables/composite.tex
@@ -1,27 +1,26 @@
-
-\begin{tabularx}{\linewidth}{|r||r||r|r||r|r|r||C|}
+\begin{expandtable}
+\begin{tabularx}{\linewidth}{|r||r|R||r||r|}
 	\hline
-	\multicolumn{8}{|c|}{Important Numbers} \\
+	$10^x$ &             Highly Composite &  \# Divs & \# prime Divs &                   \# Primes \\
 	\hline
-	$10^x$ &             Highly Composite &  \# Divs & $<$ Prime & $>$ Prime &                   \# Primes & primorial & \\
-	\hline
-	     1 &                            6 &        4 &      $-3$ &      $+1$ &                           4 &         2 & \\
-	     2 &                           60 &       12 &      $-3$ &      $+1$ &                          25 &         3 & \\
-	     3 &                          840 &       32 &      $-3$ &      $+9$ &                         168 &         4 & \\
-	     4 &                       7\,560 &       64 &     $-27$ &      $+7$ &                      1\,229 &         5 & \\
-	     5 &                      83\,160 &      128 &      $-9$ &      $+3$ &                      9\,592 &         6 & \\
-	     6 &                     720\,720 &      240 &     $-17$ &      $+3$ &                     78\,498 &         7 & \\
-	     7 &                  8\,648\,640 &      448 &      $-9$ &     $+19$ &                    664\,579 &         8 & \\
-	     8 &                 73\,513\,440 &      768 &     $-11$ &      $+7$ &                 5\,761\,455 &         8 & \\
-	     9 &                735\,134\,400 &   1\,344 &     $-63$ &      $+7$ &                50\,847\,534 &         9 & \\
-	    10 &             6\,983\,776\,800 &   2\,304 &     $-33$ &     $+19$ &               455\,052\,511 &        10 & \\
-	    11 &            97\,772\,875\,200 &   4\,032 &     $-23$ &      $+3$ &            4\,118\,054\,813 &        10 & \\
-	    12 &           963\,761\,198\,400 &   6\,720 &     $-11$ &     $+39$ &           37\,607\,912\,018 &        11 & \\
-	    13 &        9\,316\,358\,251\,200 &  10\,752 &     $-29$ &     $+37$ &          346\,065\,536\,839 &        12 & \\
-	    14 &       97\,821\,761\,637\,600 &  17\,280 &     $-27$ &     $+31$ &       3\,204\,941\,750\,802 &        12 & \\
-	    15 &      866\,421\,317\,361\,600 &  26\,880 &     $-11$ &     $+37$ &      29\,844\,570\,422\,669 &        13 & \\
-	    16 &   8\,086\,598\,962\,041\,600 &  41\,472 &     $-63$ &     $+61$ &     279\,238\,341\,033\,925 &        13 & \\
-	    17 &  74\,801\,040\,398\,884\,800 &  64\,512 &      $-3$ &      $+3$ &  2\,623\,557\,157\,654\,233 &        14 & \\
-	    18 & 897\,612\,484\,786\,617\,600 & 103\,680 &     $-11$ &      $+3$ & 24\,739\,954\,287\,740\,860 &        16 & \\
+	     1 &                            6 &        4 &             2 &                           4 \\
+	     2 &                           60 &       12 &             3 &                          25 \\
+	     3 &                          840 &       32 &             4 &                         168 \\
+	     4 &                       7\,560 &       64 &             5 &                      1\,229 \\
+	     5 &                      83\,160 &      128 &             6 &                      9\,592 \\
+	     6 &                     720\,720 &      240 &             7 &                     78\,498 \\
+	     7 &                  8\,648\,640 &      448 &             8 &                    664\,579 \\
+	     8 &                 73\,513\,440 &      768 &             8 &                 5\,761\,455 \\
+	     9 &                735\,134\,400 &   1\,344 &             9 &                50\,847\,534 \\
+	    10 &             6\,983\,776\,800 &   2\,304 &            10 &               455\,052\,511 \\
+	    11 &            97\,772\,875\,200 &   4\,032 &            10 &            4\,118\,054\,813 \\
+	    12 &           963\,761\,198\,400 &   6\,720 &            11 &           37\,607\,912\,018 \\
+	    13 &        9\,316\,358\,251\,200 &  10\,752 &            12 &          346\,065\,536\,839 \\
+	    14 &       97\,821\,761\,637\,600 &  17\,280 &            12 &       3\,204\,941\,750\,802 \\
+	    15 &      866\,421\,317\,361\,600 &  26\,880 &            13 &      29\,844\,570\,422\,669 \\
+	    16 &   8\,086\,598\,962\,041\,600 &  41\,472 &            13 &     279\,238\,341\,033\,925 \\
+	    17 &  74\,801\,040\,398\,884\,800 &  64\,512 &            14 &  2\,623\,557\,157\,654\,233 \\
+	    18 & 897\,612\,484\,786\,617\,600 & 103\,680 &            16 & 24\,739\,954\,287\,740\,860 \\
 	\hline
 \end{tabularx}
+\end{expandtable}
diff --git a/content/math/tables/nim.tex b/content/math/tables/nim.tex
index 8490d42..66e289e 100644
--- a/content/math/tables/nim.tex
+++ b/content/math/tables/nim.tex
@@ -1,7 +1,6 @@
+\begin{expandtable}
 \begin{tabularx}{\linewidth}{|p{0.37\linewidth}|X|}
 	\hline
-	\multicolumn{2}{|c|}{Nim-Spiele (\ding{182} letzter gewinnt (normal), \ding{183} letzter verliert)} \\
-	\hline
 	Beschreibung &
 	Strategie \\
 	\hline
@@ -94,3 +93,5 @@
 	Periode ab $n = 72$ der Länge $12$.\\
 	\hline
 \end{tabularx}
+\end{expandtable}
+	
\ No newline at end of file
diff --git a/content/math/tables/numbers.tex b/content/math/tables/numbers.tex
deleted file mode 100644
index 1dc9f38..0000000
--- a/content/math/tables/numbers.tex
+++ /dev/null
@@ -1,59 +0,0 @@
-\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}
diff --git a/content/math/tables/platonic.tex b/content/math/tables/platonic.tex
index f4ee554..2866ccf 100644
--- a/content/math/tables/platonic.tex
+++ b/content/math/tables/platonic.tex
@@ -1,39 +1,39 @@
+\begin{expandtable}
 \begin{tabularx}{\linewidth}{|X|CCCX|}
 	\hline
-	\multicolumn{5}{|c|}{Platonische Körper} \\
-	\hline
-	Übersicht       & Seiten & Ecken & Kanten & dual zu \\
+	Übersicht       & |F|    & |V|   & |E|    & dual zu \\
 	\hline
 	Tetraeder       & 4      & 4     & 6      & Tetraeder \\
-	Würfel/Hexaeder & 6      & 8     & 12     & Oktaeder \\
-	Oktaeder        & 8      & 6     & 12     & Würfel/Hexaeder\\
+	Würfel          & 6      & 8     & 12     & Oktaeder \\
+	Oktaeder        & 8      & 6     & 12     & Würfel\\
 	Dodekaeder      & 12     & 20    & 30     & Ikosaeder \\
 	Ikosaeder       & 20     & 12    & 30     & Dodekaeder \\
 	\hline
 	\multicolumn{5}{|c|}{Färbungen mit maximal $n$ Farben (bis auf Isomorphie)} \\
 	\hline
-	\multicolumn{3}{|l}{Ecken vom Oktaeder/Seiten vom Würfel} &
+	\multicolumn{3}{|l}{|V| vom Oktaeder/|F| vom Würfel} &
 	\multicolumn{2}{l|}{$(n^6 + 3n^4 + 12n^3 + 8n^2)/24$} \\
 
-	\multicolumn{3}{|l}{Ecken vom Würfel/Seiten vom Oktaeder} &
+	\multicolumn{3}{|l}{|V| vom Würfel/|F| vom Oktaeder} &
 	\multicolumn{2}{l|}{$(n^8 + 17n^4 + 6n^2)/24$} \\
 
-	\multicolumn{3}{|l}{Kanten vom Würfel/Oktaeder} &
+	\multicolumn{3}{|l}{|E| vom Würfel/Oktaeder} &
 	\multicolumn{2}{l|}{$(n^{12} + 6n^7 + 3n^6 + 8n^4 + 6n^3)/24$} \\
 
-	\multicolumn{3}{|l}{Ecken/Seiten vom Tetraeder} &
+	\multicolumn{3}{|l}{|V|/|F| vom Tetraeder} &
 	\multicolumn{2}{l|}{$(n^4 + 11n^2)/12$} \\
 
-	\multicolumn{3}{|l}{Kanten vom Tetraeder} &
+	\multicolumn{3}{|l}{|E| vom Tetraeder} &
 	\multicolumn{2}{l|}{$(n^6 + 3n^4 + 8n^2)/12$} \\
 
-	\multicolumn{3}{|l}{Ecken vom Ikosaeder/Seiten vom Dodekaeder} &
+	\multicolumn{3}{|l}{|V| vom Ikosaeder/|F| vom Dodekaeder} &
 	\multicolumn{2}{l|}{$(n^{12} + 15n^6 + 44n^4)/60$} \\
 
-	\multicolumn{3}{|l}{Ecken vom Dodekaeder/Seiten vom Ikosaeder} &
+	\multicolumn{3}{|l}{|V| vom Dodekaeder/|F| vom Ikosaeder} &
 	\multicolumn{2}{l|}{$(n^{20} + 15n^{10} + 20n^8 + 24n^4)/60$} \\
 
-	\multicolumn{3}{|l}{Kanten vom Dodekaeder/Ikosaeder (evtl. falsch)} &
+	\multicolumn{3}{|l}{|E| vom Dodekaeder/Ikosaeder} &
 	\multicolumn{2}{l|}{$(n^{30} + 15n^{16} + 20n^{10} + 24n^6)/60$} \\
 	\hline
 \end{tabularx}
+\end{expandtable}
diff --git a/content/math/tables/probability.tex b/content/math/tables/probability.tex
index f265d10..29f92e1 100644
--- a/content/math/tables/probability.tex
+++ b/content/math/tables/probability.tex
@@ -1,19 +1,15 @@
-\begin{tabularx}{\linewidth}{|LICIR|}
+\begin{expandtable}
+\begin{tabularx}{\linewidth}{|LIR|}
 	\hline
-	\multicolumn{3}{|c|}{
+	\multicolumn{2}{|c|}{
 		Wahrscheinlichkeitstheorie ($A,B$ Ereignisse und $X,Y$ Variablen)
 	} \\
 	\hline
-	$\E(X + Y) = \E(X) + \E(Y)$ &
-	$\E(\alpha X) = \alpha \E(X)$ &
-	$X, Y$ unabh. $\Leftrightarrow \E(XY) = \E(X) \cdot \E(Y)$\\
-	
-	$\Pr[A \vert B] = \frac{\Pr[A \land B]}{\Pr[B]}$ &
-	$A, B$ disj. $\Leftrightarrow \Pr[A \land B] = \Pr[A] \cdot \Pr[B]$ &
-	$\Pr[A \lor B] = \Pr[A] + \Pr[B] - \Pr[A \land B]$ \\
+	$\E(X + Y) = \E(X) + \E(Y)$                                & $\Pr[A \vert B] = \frac{\Pr[A \land B]}{\Pr[B]}$ \\
+	$\E(\alpha X) = \alpha \E(X)$                              & $\Pr[A \lor B] = \Pr[A] + \Pr[B] - \Pr[A \land B]$ \\
+	$X, Y$ unabh. $\Leftrightarrow \E(XY) = \E(X) \cdot \E(Y)$ & $A, B$ disj. $\Leftrightarrow \Pr[A \land B] = \Pr[A] \cdot \Pr[B]$\\
 	\hline
 \end{tabularx}
-\vfill
 \begin{tabularx}{\linewidth}{|Xlr|lrX|}
 	\hline
 	\multicolumn{6}{|c|}{\textsc{Bertrand}'s Ballot Theorem (Kandidaten $A$ und $B$, $k \in \mathbb{N}$)} \\
@@ -25,3 +21,4 @@
 	$\#A \geq \#B + k$ & $Num = \frac{a - k + 1 - b}{a - k + 1} \binom{a + b - k}{b}$ & \\
 	\hline
 \end{tabularx}
+\end{expandtable}
diff --git a/content/math/tables/series.tex b/content/math/tables/series.tex
index 3042781..9618c2b 100644
--- a/content/math/tables/series.tex
+++ b/content/math/tables/series.tex
@@ -1,33 +1,33 @@
-\begin{tabularx}{\linewidth}{|XIXIXIX|}
-	\hline
-	\multicolumn{4}{|c|}{Reihen} \\
+\begin{expandtable}
+\begin{tabularx}{\linewidth}{|XIXIX|}
 	\hline
 	$\sum\limits_{i = 1}^n i = \frac{n(n+1)}{2}$ &
 	$\sum\limits_{i = 1}^n i^2 = \frac{n(n + 1)(2n + 1)}{6}$ &
-	$\sum\limits_{i = 1}^n i^3 = \frac{n^2 (n + 1)^2}{4}$ &
-	$H_n = \sum\limits_{i = 1}^n \frac{1}{i}$ \\
+	$\sum\limits_{i = 1}^n i^3 = \frac{n^2 (n + 1)^2}{4}$ \\
 	\grayhline
 	
-	$\sum\limits_{i = 0}^n c^i = \frac{c^{n + 1} - 1}{c - 1} \quad c \neq 1$ &	
-	$\sum\limits_{i = 0}^\infty c^i = \frac{1}{1 - c} \quad \vert c \vert < 1$ &
-	$\sum\limits_{i = 1}^\infty c^i = \frac{c}{1 - c} \quad \vert c \vert < 1$ &	
-	$\sum\limits_{i = 0}^\infty ic^i = \frac{c}{(1 - c)^2} \quad \vert c \vert < 1$ \\
+	$\sum\limits_{i = 0}^n c^i = \frac{c^{n + 1} - 1}{c - 1} \hfill c \neq 1$ &	
+	$\sum\limits_{i = 0}^\infty c^i = \frac{1}{1 - c} \hfill \vert c \vert < 1$ &
+	$\sum\limits_{i = 1}^\infty c^i = \frac{c}{1 - c} \hfill \vert c \vert < 1$ \\
 	\grayhline
-	
+		
 	\multicolumn{2}{|lI}{
 		$\sum\limits_{i = 0}^n ic^i = \frac{nc^{n + 2} - (n + 1)c^{n + 1} + c}{(c - 1)^2} \quad c \neq 1$
 	} &	
-	\multicolumn{2}{l|}{
+	$\sum\limits_{i = 0}^\infty ic^i = \frac{c}{(1 - c)^2} \hfill \vert c \vert < 1$ \\
+	\grayhline	
+	
+	\multicolumn{2}{|lI}{
 		$\sum\limits_{i = 1}^n iH_i = \frac{n(n + 1)}{2}H_n - \frac{n(n - 1)}{4}$
-	} \\
+	} &
+	$H_n = \sum\limits_{i = 1}^n \frac{1}{i}$ \\
 	\grayhline
 	
 	\multicolumn{2}{|lI}{
-		$\sum\limits_{i = 1}^n H_i = (n + 1)H_n - n$
-	} &
-	\multicolumn{2}{l|}{
 		$\sum\limits_{i = 1}^n \binom{i}{m}H_i =
 		\binom{n + 1}{m + 1} \left(H_{n + 1} - \frac{1}{m  + 1}\right)$
-	} \\
+	} &
+	$\sum\limits_{i = 1}^n H_i = (n + 1)H_n - n$ \\
 	\hline
 \end{tabularx}
+\end{expandtable}
diff --git a/content/math/tables/stuff.tex b/content/math/tables/stuff.tex
index 3cf8b4c..82f2d3f 100644
--- a/content/math/tables/stuff.tex
+++ b/content/math/tables/stuff.tex
@@ -1,6 +1,7 @@
-\begin{tabularx}{\linewidth}{|ll|}
+\begin{expandtable}
+\begin{tabularx}{\linewidth}{|Ll|}
 	\hline
-	\multicolumn{2}{|C|}{Verschiedenes} \\
+	\multicolumn{2}{|c|}{Verschiedenes} \\
 	\hline
 	Türme von Hanoi, minimale Schirttzahl: &
 	$T_n = 2^n - 1$ \\
@@ -20,7 +21,7 @@
 	\#Wälder mit $k$ gewurzelten Bäumen	&
 	$\frac{k}{n}\binom{n}{k}n^{n-k}$ \\
 
-	\#Wälder mit $k$ gewurzelten Bäumen mit vorgegebenen Wurzelknoten&
+	\#Wälder mit $k$ gewurzelten~Bäumen mit vorgegebenen Wurzelknoten&
 	$\frac{k}{n}n^{n-k}$ \\
 
 	Derangements &
@@ -29,4 +30,4 @@
 	$\lim\limits_{n \to \infty} \frac{!n}{n!} = \frac{1}{e}$ \\
 	\hline
 \end{tabularx}
-
+\end{expandtable}