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{\small
{\bf Odhady:}
\begin{eqnarray*}
H_n &=& 
  \ln{n} +\gamma +{1 \over 2n}+{1 \over 12n^2}+
         {1 \over  120n^4}+O(n^{-6})
\\
n! &=& \sqrt{2\pi n} \left({n \over e}\right)^n
  \left(1+ {1 \over 12n} + {1\over 288n^2}
  - { 139 \over 51840 n^3} + O\left(n^{-4}\right) \right)
\\
\Pi (n) &=& {n \over \ln{n}} + {n \over \ln^2{n} } + { 2! \over \ln^3{n} }
+ {3! \over \ln^4{n} } + O({n \over \log^5{n}}) 
\\
e^z &=& 1 + z + {z^2 \over 2! } + {z^3 \over 3!} + {z^4 \over 4! } +
O(z^5)
\\
 \ln{(1+z)} &=& z - {z^2 \over 2 } + {z^3 \over 3} - {z^4 \over 4} + O(z^5)
\\
 {1 \over 1-z} &=& 1 + z + z^2 + z^3 + z^4 + O(z^5)
\\
 (1+z)^\alpha &=& 1 + \alpha z + {\alpha \choose 2} z^2 + {\alpha \choose 3}
z^3 + {\alpha \choose 4} z^4 + O(z^5)
\end{eqnarray*}

}
{\small
{\bf Manipulacia s generujucimi funkciami:}
\begin{eqnarray*}
\alpha F(z) + \beta G(z) &=& \sum_n (\alpha f_n + \beta g_n) z^n
\\
z^m G(z) &=& \sum_n g_{n-m} z^n ,\qquad {\rm integer}\  m\geq 0
\\
{ G(z) - g_0 - g_1 z - \dots - g_{m-1} z^{m-1} \over z^m} &=& \sum_{n \geq
0} g_{n+m} z^n , \qquad {\rm integer}\  m\geq 0
\\
G(cz) &=& \sum_n c^n g_n z^n
\\
G'(z) &=& \sum_n (n+1)g_{n+1}z^n
\\
zG'(z) &=& \sum_n ng_nz^n
\\
\int_0^z G(t) dt &=& \sum_{n \geq 1} {1 \over n} g_{n-1} z^n
\\
F(z)G(z) &=& \sum_n \left( \sum_k f_k g_{n-k} \right)z^n
\\
{1 \over 1-z} G(z) &=& \sum_n \left( \sum_{k\leq n} g_k \right) z^n
\\
\end{eqnarray*}
}
{\small
{\bf Fibonacci:}
\begin{eqnarray*}
F_{2n+2} &=& F_1 + F_3 + \dots + F_{2n+1} \\
F_{2n+1} &=& F_0 + F_2 + F_4 + \dots + F_{2n} \\
F_n^2 - F_{n+1} F_{n-1} &=& (-1)^{n+1} \\
F_n F_{n+1} &=& F_1^2 + F_2^2 + \dots F_n^2 \\
F_{2n}^2 &=& F_1F_2 + F_2F_3 + \dots F_{2n-1}F_{2n} \\
F_{n+2} &=& F_0 + F_1 + F_2 + \dots + F_n \\
F_{n+m} &=& F_{m-1}F_n + F_mF_{n+1} \\
F_{n+1}F_{n+2} - F_nF_{n+3} &=& (-1)^n \\
F_{2n-1} &=& F_n^2 + F_{n-1}^2 \\
F_{3n} &=& F_{n+1}^3 + F_n^3 - F_{n-1} ^3 \\
F_n^4 - F_{n-2}F_{n-1}F_{n+1}F_{n+2} &=& 1 \\
\end{eqnarray*}
}
{\small
{\bf Jednoduche postupnosti a ich generujuce funkcie:}
\begin{eqnarray*}
{\rm sequence} & {\rm generating\ funcion} & {\rm closed\ form}
\\
\langle 1,0,0,0,0,0,\dots \rangle & \sum_{n\geq 0} [n=0]z^n & 1
\\
\langle 0,\dots ,0,1,0,0,\dots \rangle & \sum_{n\geq 0} [n=m]z^n & z^m
\\
\langle 1,1,1,1,1,1,\dots \rangle & \sum_{n\geq 0} z^n & {1 \over 1-z}
\\
\langle 1, -1, 1, -1, 1, -1, \dots \rangle & \sum_{n\geq 0}(-1)^n z^n & {1
\over 1+z}
\\
\langle 1,0,1,0,1,0, \dots \rangle & \sum_{n\geq 0} [2\backslash n] z^n & {1 \over
1-z^2}
\\
\langle 1,0,\dots ,0, 1,0,\dots ,0, 1, 0, \dots \rangle & \sum_{n \geq 0}
[m\backslash n]z^n & {1 \over 1-z^m}
\\
\langle 1,2,3,4,5,6, \dots \rangle & \sum_{n \geq 0}(n+1)z^n & {1 \over (1-z)^2}
\\
\langle 1,2,4,8,16,32, \dots \rangle & \sum_{n \geq 0}2^n z^n & {1 \over
1-2z}
\\
\langle 1,4,6,4,1,0,0,\dots \rangle & \sum_{n \geq 0} {4 \choose n} z^n &
(1+z)^4
\\
\langle 1,c, {c \choose 2}, {c \choose 3}, \dots \rangle & \sum_{n\geq 0} {c
\choose n} z^n & (1+z)^c
\\
\langle 1, c, {c+1 \choose 2}, {c+2 \choose 3}, \dots \rangle & \sum_{n \geq
0}{c+n-1 \choose n} z^n & {1 \over (1-z)^c}
\\
\langle 1, c, c^2, c^3, \dots \rangle & \sum_{n \geq 0} c^nz^n & {1 \over
1-cz}
\\
\langle 1, {m+1 \choose m}, {m+2 \choose m}, {m+3 \choose m}, \dots \rangle &
\sum_{n \geq 0} {m+n \choose m} z^n & {1 \over (1-z)^{m+1} }
\\
\langle 0,1,{1\over 2}, {1 \over 3}, {1 \over 4}, \dots \rangle & \sum_{n
\geq 1} {1\over n} z^n & \ln{\left({1 \over 1-z}\right)}
\\
\langle 0,1,-{1\over 2}, {1 \over 3}, -{1 \over 4}, \dots \rangle & \sum_{n
\geq 1} {(-1)^{n+1} \over n} z^n & \ln{(1+z)}
\\
\langle 1,1,{1\over 2}, {1 \over 6}, {1 \over 24}, {1 \over 120}, \dots
\rangle & \sum_{n \geq 0} {1 \over n!} z^n & e^z
\\
\end{eqnarray*}
}
{\small
{\bf Ine vzorceky:}
\begin{eqnarray*}
\sum_{k=0}^n a^k &=& {a^{n+1} -1 \over a-1 } \\
\sum_{k=0}^n {1 \over a^k} &=& {a \over a-1} - {1 \over (a-1)a^n } \\
\sum_{k=0}^n k\cdot a^k &=& { na^{n+2} - (n+1)a^{n+1} + a \over (a-1)^2 } \\
k \cdot k! &=& (k+1)! - k! \\
H_k^{(m)} &=& \sum_{k=1}^n {1 \over k^m} \\
\sum_{k=1}^{n-1} H_k &=& nH_n -n \\
\sum_{k=0}^n k &=& { n(n+1) \over 2 } \\
\sum_{k=0}^n k^2 &=& {n(n+1)(2n+1) \over 6} \\
\sum_{k=0}^n k^3 &=& {n \choose 2 }^2 \\
(n+1) H_{n+1} &=& nH_n +1 \\
{n \choose k} &=& {n \choose n-k} \\
{r \choose k} &=& {r \over k} {r-1 \choose k-1} \\
(r-k) {r \choose k} &=& r {r-1 \choose k} \\
\end{eqnarray*}
}
\end{document}
