1
vote
2answers
59 views

Find a closed form for the generating function for this sequence

The sequence: $0, 0, 0, 1, 1, 1, 1, 1, 1, \ldots$ The book gives the answer of $\frac{x^3}{1-x}$ but I'm not sure how to get this answer. I understand the generating function of this sequence will be ...
4
votes
1answer
73 views

Closed form of $\sum_{k=0}^nk\binom{k}{3}\binom{2n}{k}$

Recently, I came across the following exercise on the course of discrete math Find a closed form for $\sum_{k=0}^nk\binom{k}{3}\binom{2n}{k}$ So I tried some of the usual techniques: Let ...
8
votes
3answers
227 views

Harmonic Numbers series I

Can it be shown that \begin{align} \sum_{n=1}^{\infty} \binom{2n}{n} \ \frac{H_{n+1}}{n+1} \ \left(\frac{3}{16}\right)^{n} = \frac{5}{3} + \frac{8}{3} \ \ln 2 - \frac{8}{3} \ \ln 3 \end{align} where ...
0
votes
0answers
39 views

Are generating functions ever analytic for logarithmic series?

Given a series $s_n = \ln(n) f(n)$ where $f(\cdot)$ is an elementary analytic function which does not involve the logarithm. More precisely $f$ can have simple poles but no branch cuts or essential ...
0
votes
1answer
61 views

Closed form for nth term of generating function

How would I find the closed form for the $n^{th}$ term of a sequence? Is there a general formula I can follow for these types of problems? Taking this sequence for example... $$\frac{x^5}{(1-x)^4}$$
5
votes
2answers
73 views

Closed form of generating function consisting of power of two binomials

Let $g(x)$ be infinite formal power series and $$g(x) = (1 + x)(1 + x^2)\cdots(1 + x^{2^k})\cdots$$ Show that $(1 - x) g(x) = 1$. My book gives following proof: Using a fact that $(1 - x^k)(1 + ...
0
votes
0answers
66 views

Is there a way to simplify Legendre-squared sum $\sum_{n} \frac{[P'_{n}(x)]^2}{n(n+1)}$

Is there a closed-form expression for $$F(x) = \sum_{n=2,{\rm even}}^{\infty} \frac{[P'_{n}(x)]^2}{n(n+1)}$$ where $P'_{n}(x)$ is the derivative of the $n^{\rm th}$ Legendre polynomial? Simple ...
2
votes
0answers
60 views

Find the sum of exponentails of squares $\sum_{r=1}^n e^{-\alpha r^2}$

I would like to find $$a_n =\sum_{r=1}^n e^{-\alpha r^2},\qquad \alpha\in\mathbb{R}$$ I tried to solve the equivalent recursion $$a_n=a_{n-1}+e^{-\alpha n^2}\quad(n>0),\qquad a_0=0.$$ with an ...
2
votes
1answer
59 views

Generating function of $ \lim_{x\rightarrow 0} \frac{1}{n!} \frac{\partial^n}{\partial x^n} [(1+ax)^n f(x)] $

Is there a closed form for the generating function (or exponential generating function) of the sequence $$s_n= \lim_{x\rightarrow 0}\frac{1}{n!} \frac{\partial^n}{\partial x^n}\left[(1+ax)^n f(x) ...
2
votes
1answer
96 views

Evaluate $\sum_{n=1}^{\infty} \frac{\ln n}{(n+1)!}$

Is there a closed form expression for this sum (or very similar sums) with known transcendental functions or irrational numbers? $$\sum_{n=1}^{\infty} \frac{\ln n}{(n+1)!} \approx ...
1
vote
3answers
221 views

Is there a closed form for the sum $\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$?

I am interested in finding a closed form for the sum $\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$. Does anyone know if there is some Binomial identity that might be helpful here? Thank you.
3
votes
4answers
109 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
569 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 = ...
4
votes
2answers
237 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
178 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
556 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
358 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 $\displaystyle{\frac{(1-2x)}{(1+3x)^3}}$ = ...