Question

Consider the sequence {fn}nzo with recurrence given by fo = 1 and п ("+")s.-6 in > 1 i=0 Find its exponential generating function E(2).

Answer 1

ANSWER :

sequence $\left \{ f_{n} \right \}_{n\geq 0}$ with recurrence given by and

$\sum_{i=0}^{n}\begin{pmatrix} n+1 \\ i \end{pmatrix}f_{i}=0,n\geq 1$

For i=0, $\sum_{i=0}^{n}\begin{pmatrix} n+1 \\ 0 \end{pmatrix}(1)=\sum_{i=0}^{n}\begin{pmatrix} n+1 \\ 0 \end{pmatrix}$

For generating function for a squence

$\left \{ f_{n} \right \}_{n\geq 0}\Rightarrow E(x)=\sum_{n=0}^{\infty }\frac{f_{n}}{n!}x^{n}.......(1)$

$\therefore \sum_{i=0}^{0 }\begin{pmatrix} n+1 \\ i \end{pmatrix}f_{i}=(n+1)!.......(2)$

So, exponential generating function for all permutations is

$=\sum_{n=0}^{\infty }\begin{pmatrix} n+1 \\ i \end{pmatrix}(n+1)!\left ( \frac{x^{n}}{n!} \right )=E(x).......(2)$