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\centerline{HOMEWORK SET \#2 / CO1A /  Spring 2020}  \vskip3mm
\item{1.} Prove that $\sum_{k=1}^n {(-1)^{k-1} \over k}{n\choose
k}=1+{1\over 2}+{1\over3}+ \cdots + {1\over n}$.
\item{2.} Show (preferably by a combinatorial argument) that $
S(n,k)=kS(n-1,k)+S(n-1,k-1).$
\item{3.}   Show by a combinatorial argument that\vskip1mm
\centerline{$S(n+1,k)={n \choose 0}S(0,k-1)+
{n \choose 1}S(1,k-1)+\cdots+{n\choose n}
S(n,k-1).$}
\item{4.} Compute $S(n,4)$, S$(n, n-3)$ and S$(n, n-4)$.
\item{5.} In how many different ways can you reach the bottom-right
corner of a chessboard from the top-left corner, if you may only
step right and downward and you may never be above the top-left
bottom-right diagonal of the board (hint: think of the Pascal triangle)
\item{6.} Find the ordinary generating function of the sequence
$n^2$, i.e. find a close formula for the function
$\sum_{n=0}^\infty n^2\cdot x^n$.
\item{7.} Find the sequence $(a_k)$ such that the function
${1\over 1 + x^4}$ is the ordinary generating function of
$(a_k)$, that is ${1\over 1 + x^4}=\sum_{n=0}^\infty a_n\cdot x^n$.
\item{8.} Find the sequence $(b_k)$ such that the function
${b\over a + x}$ is the ordinary generating function of $(b_k)$,
that is ${b\over a + x}=\sum_{n=0}^\infty b_n\cdot x^n$.



 \vfill \vfill


\centerline{HOMEWORK SET \#2 / CO1A /  Spring 2020}  \vskip3mm
\item{1.} Prove that $\sum_{k=1}^n {(-1)^{k-1} \over k}{n\choose
k}=1+{1\over 2}+{1\over3}+ \cdots + {1\over n}$.
\item{2.} Show (preferably by a combinatorial argument) that $
S(n,k)=kS(n-1,k)+S(n-1,k-1).$
\item{3.}   Show by a combinatorial argument that\vskip1mm
\centerline{$S(n+1,k)={n \choose 0}S(0,k-1)+
{n \choose 1}S(1,k-1)+\cdots+{n\choose n}
S(n,k-1).$}
\item{4.} Compute $S(n,4)$, S$(n, n-3)$ and S$(n, n-4)$.
\item{5.} In how many different ways can you reach the bottom-right
corner of a chessboard from the top-left corner, if you may only
step right and downward and you may never be above the top-left
bottom-right diagonal of the board (hint: think of the Pascal triangle)
\item{6.} Find the ordinary generating function of the sequence
$n^2$, i.e. find a close formula for the function
$\sum_{n=0}^\infty n^2\cdot x^n$.
\item{7.} Find the sequence $(a_k)$ such that the function
${1\over 1 + x^4}$ is the ordinary generating function of
$(a_k)$, that is ${1\over 1 + x^4}=\sum_{n=0}^\infty a_n\cdot x^n$.
\item{8.} Find the sequence $(b_k)$ such that the function
${b\over a + x}$ is the ordinary generating function of $(b_k)$,
that is ${b\over a + x}=\sum_{n=0}^\infty b_n\cdot x^n$.
\bye
