Enumerative Combinatorics: Activities and Exercises

\begin{equation*} \begin{array}{rcccl} \displaystyle{ (1+x)^k } &=& \displaystyle{ \sum_{n=0}^{k} {k \choose n} x^n }&=& \displaystyle{ 1 + {k \choose 1} x + {k \choose 2} x^2 + {k \choose 3} x^3 + \cdots + {k \choose k} x^k } \\ \displaystyle{ (1+ax)^k } &=& \displaystyle{ \sum_{n=0}^{k} {k \choose n} a^n x^n }&=& \displaystyle{ 1 + {k \choose 1} a x + {k \choose 2} a^2 x^2 + {k \choose 3} a^3 x^3 + \cdots + {k \choose k} a^k x^k } \\ \displaystyle{ (1+x^r)^k } &=& \displaystyle{ \sum_{n=0}^{k} {k \choose n} x^{nr} }&=& \displaystyle{ 1 + {k \choose 1} x^r + {k \choose 2} x^{2r} + {k \choose 3} x^{3r} + \cdots + {k \choose k} x^{kr} } \\ \end{array} \end{equation*}

\begin{equation*} \begin{array}{rcccl} \displaystyle{ (1+x^r)^a } &=& \displaystyle{ \sum_{n=0}^{\infty} {a \choose n} x^{nr} }&=& \displaystyle{ 1 + {a \choose 1} x^r + {a \choose 2} x^{2r} + {a \choose 3} x^{3r} + \cdots } \\ \end{array} \end{equation*}

\begin{equation*} \begin{array}{rcccl} \displaystyle{ \frac{1}{1-x} } &=& \displaystyle{ \sum_{n=0}^{\infty} x^n }&=& \displaystyle{ 1 + x + x^2 + x^3 + \cdots } \\ \displaystyle{ \frac{1}{1-a x} } &=& \displaystyle{ \sum_{n=0}^{\infty} a^n x^n }&=& \displaystyle{ 1 + a x + a^2 x^2 + a^3 x^3 + \cdots } \\ \displaystyle{ \frac{1}{1-x^k} } &=& \displaystyle{ \sum_{n=0}^{\infty} x^{nk} }&=& \displaystyle{ 1 + x^k + x^{2k} + x^{3k} + \cdots } \\ \displaystyle{ \frac{1}{(1-x)^2} } &=& \displaystyle{ \sum_{n=0}^{\infty} (n+1) x^{n} }&=& \displaystyle{ 1 + 2x + 3x^{2} + 4x^{3} + \cdots } \\ \displaystyle{ \frac{1}{(1-x)^k} } &=& \displaystyle{ \sum_{n=0}^{\infty} {k+n-1 \choose n} x^{n} }&=& \displaystyle{ 1 + {k \choose 1} x + {k+1 \choose 2}x^{2} + {k+2 \choose 3}x^{3} + \cdots } \\ \displaystyle{ \frac{1}{(1-ax)^k} } &=& \displaystyle{ \sum_{n=0}^{\infty} {k+n-1 \choose n} a^n x^{n} }&=& \displaystyle{ 1 + {k \choose 1} a x + {k+1 \choose 2}a^2 x^{2} + {k+2 \choose 3} a^3 x^{3} + \cdots } \\ \\ \displaystyle{ \log \left(\frac{1}{1-x} \right) } &=& \displaystyle{ \sum_{n=1}^{\infty} \frac{x^{n}}{n} }&=& \displaystyle{ x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + \frac{x^{4}}{4} +\cdots } \\ \end{array} \end{equation*}

\begin{equation*} \begin{array}{rcccl} \displaystyle{ e^x } &=& \displaystyle{ \sum_{n=0}^{\infty} \frac{x^{n}}{n!} }&=& \displaystyle{ 1+x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \frac{x^{4}}{24} +\cdots } \\ \displaystyle{ e^{ax} } &=& \displaystyle{ \sum_{n=0}^{\infty} \frac{a^n x^{n}}{n!} }&=& \displaystyle{ 1+ax + \frac{a^2x^{2}}{2} + \frac{a^3x^{3}}{6} + \frac{a^4x^{4}}{24} +\cdots } \\ \displaystyle{ e^{x^k} } &=& \displaystyle{ \sum_{n=0}^{\infty} \frac{x^{nk}}{n!} }&=& \displaystyle{ 1+x^k + \frac{x^{2k}}{2} + \frac{x^{3k}}{6} + \frac{x^{4k}}{24} +\cdots } \\ \displaystyle{ \sinh x = \frac{e^x - e^{-x}}{2} } &=& \displaystyle{ \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} }&=& \displaystyle{ x + \frac{x^{3}}{6} + \frac{x^{5}}{120} + \frac{x^{7}}{5040} +\cdots } \\ \displaystyle{ \cosh x = \frac{e^x + e^{-x}}{2} } &=& \displaystyle{ \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!} }&=& \displaystyle{ 1+ \frac{x^{2}}{2} + \frac{x^{4}}{24} + \frac{x^{6}}{720} +\cdots } \\ \end{array} \end{equation*}

\begin{equation*} \begin{array}{rcccl} \displaystyle{ \tan^{-1} x } &=& \displaystyle{ \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} }&=& \displaystyle{ x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} +\cdots } \\ \\ \displaystyle{ \log \left( \sqrt{\frac{1+x}{1-x}} \right) } &=& \displaystyle{ \sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1} }&=& \displaystyle{ x + \frac{x^{3}}{3} + \frac{x^{5}}{5} + \frac{x^{7}}{7} +\cdots } \\ \\ \displaystyle{ \frac{1}{\sqrt{1-4x}} } &=& \displaystyle{ \sum_{n=0}^{\infty} {2n \choose n} x^n }&=& \displaystyle{ 1 + 2x + 6x^2 + 20x^3 +\cdots } \\ \displaystyle{ \frac{1}{\sqrt{1-x^2}} } &=& \displaystyle{ \sum_{n=0}^{\infty} {2n \choose n} \frac{ x^{2n}}{2^{2n}} }&=& \displaystyle{ 1 + \frac{x^2}{2} + \frac{3 x^4}{8} + \frac{5x^6}{16} +\cdots } \\ \\ \displaystyle{ \frac{1-\sqrt{1-4x}}{2x} } &=& \displaystyle{ \sum_{n=0}^{\infty} \frac{1}{2n+1}{2n \choose n} x^n }&=& \displaystyle{ 1 + x + 5x^2 + 14x^3 +\cdots } \\ \end{array} \end{equation*}