Tagged Questions
3
votes
3answers
38 views
Closed form of a recurrence relation using generating functions
It's been awhile since I have done this. The sequence is $\displaystyle a_n = a_{n-1} + 5~a_{n-2}$ with $a_{0}=0$ and $a_{1}=1$.
I found the generating function to be $\displaystyle G(x) = ...
-2
votes
1answer
173 views
Find a closed form for a generating function and recurrence
Find a closed form for the generating function $R(x) = \sum_{n=0}^\infty r_nx^n$, where $r_n$ is given by the recurrence $r_n = 3r_{n-1} + 5r_{n-2} + 6n$
for $n \geq 2$ and initial conditions $r_0 = ...
5
votes
2answers
88 views
Is there a closed form expression for the first half of the Binomial series?
I'm looking for a closed form expression for the sum
$P_n(x) =\sum_{0\leq k\leq n/2}\binom{n}{k}x^k$,
where $n$ is a given positive integer and $k$ runs over nonnegative integers between $0$ and ...
2
votes
0answers
119 views
What is the closed form of generating function of a power law?
I want to know if there is a "closed form" of the following generating function,
$G_n(x) = \sum_{n=0}^{\infty} P_n x^n$
where,
$P_n = C(n_0 + n)^{-\gamma}$
where $C$ is a normalization constant, ...
3
votes
3answers
374 views
How to get closed form from generating function?
I have this generating function:
$$\frac{1}{2}\, \left( {\frac {1}{\sqrt {1-4\,z}}}-1 \right) \left( \,{
\frac {1-\sqrt {1-4\,z}}{2z}}-1 \right)$$
and I know that $\frac {1}{\sqrt {1-4\,z}}$ is ...
2
votes
2answers
262 views
Generating Function explicit formula
Say i got
$\displaystyle{\frac{(1-2x)}{(1+3x)^3}}$
I used $\displaystyle{\frac{1}{(1+3x)}}$ $=\sum_{n=0}^\infty(-3)^n x^n$ and differentiated twice
I got $\displaystyle{\frac{(1-2x)}{(1+3x)^3}}$ = ...
