0
votes
0answers
15 views

Difficult generating function

Prove that the coefficient of $x^i$ for $i=1,2,3,...$ in the expansion of $\prod_{n=0}^{\infty} (1+x^{3^n})(1+x^{4^n})(1+x^{5^n})(1+x^{6^n})$ is greater than $0$.
-1
votes
1answer
60 views

Fermat Last theorem on Poly-Euler numbers

The poly-Euler numbers, denoted as $E_{n}^{(k)}$, are defined by the following generating functions :$${2\operatorname{Li}_k(1-e^{-x}) \over 1+e^{-x}}=\sum_{n=0}^\infty E_n^{(k)}{x^n\over n!}$$ The ...
0
votes
1answer
33 views

Exponential generating function for number of 10 length sequences built from the alphabet, with some restrictions

I've got the following homework question. If anybody could possibly point me in the right direction, that would be great: Suppose X is a sequence with 10 terms built from 26 letters {a, b, c, ..., ...
5
votes
2answers
288 views

Applications of generating functions to number theory

I am familiar (at least at a cursory level) with the extensive role generating functions play in the theory of partitions. What are some other prominent applications of generating functions to number ...
1
vote
1answer
61 views

Proof that the series for the generating function of the partition function converges?

For $|q| < 1$, the generating function of the partition function $p(n)$ is given by $$ \sum_{n=0}^\infty p(n) q^n = \prod_{k=1}^\infty {1 \over 1-q^k}. \tag{1} $$ I have an intuitive ...
1
vote
1answer
75 views

Ramanujan Notebook Part 1 (1.16): $\sum q^{n^2} = (-q;q^2)_\infty^2(q^2;q^2)_\infty=\frac{(-q;-q)_\infty}{(q;-q)_\infty}$

I am having trouble with proving a statement in Ramanujan's Lost Notebook Part 1 (1.16). The statement is as follows: $\varphi(q)=f(q,q)=\sum_{n=-\infty}^\infty q^{n^2} = ...
3
votes
2answers
123 views

Generating function for the characteristic function of primes

What do we know about the generating function of $\chi(n)$ (A010051) $$ f(x) = \sum_{n=0}^\infty \chi(n)x^n = \sum_{p\text{ prime}} x^p $$ for $\chi(n)$ the characteristic function of the primes: ...
0
votes
0answers
42 views

Prove an equation

Let $F(n,k)=\frac{n}{n-1}F(n-1,k)+\frac{n}2F(n-1,k-1)$, and $F(n,0)=1, F(n,k)=0$( if $n\le k+2$). Is the equation $F(n,k)=\frac{{\binom{n-3}{k}}{\binom{n+k-1}{k}}}{k+1}$ right? If it is right, please ...
0
votes
0answers
38 views

The Generating Functions for $p(S_d,n)$

Prove: $$\sum_{n=0}^{\infty}p(S_2,n)q^n=\prod_{j=1}^{\infty}\frac{1}{(1-q^{5j-4})(1-q^{5j-1})}$$ I have been trying to figure out where to even start this problem and I have no idea what to do. Can ...
1
vote
1answer
64 views

Finding the number of integral solutions to an equation given constraints.

I'm having a huge problem even understanding what to do here. Any guidance to get me going would be greatly appreciated. If $n,k$ are positive integers, how many integral solutions are there to the ...
0
votes
0answers
68 views

The number of compositions of n into exactly k parts excluding inversions

The number of compositions of n into exactly k parts is well known. I would like to know this number excluding inversions i.e. wherein (1+2+3+4) and (4+3+2+1) is the same composition.
0
votes
1answer
42 views

Number of partitions of integer into parts repeated <= 2 times

The generating function for the number of such partitions is $$ G(q) = \prod_{i=0}^{\infty}(1+q^i+q^{2i}) $$ - that much I understand. Is there any way to transform it into a form ...
5
votes
4answers
240 views

Check whether $\sum\limits_{n=1}^{\infty}\frac{z^n}{(1-z^n)^k}=\sum\limits_{n=1}^{\infty}\sigma_{k-1}(n)z^n$

Is it true that $$\sum_{n=1}^{\infty}\frac{z^n}{(1-z^n)^k}=\sum_{n=1}^{\infty}\sigma_{k-1}(n)z^n$$ If yes, how can I prove it?
3
votes
2answers
83 views

For what $n$ is it true that $(1+\sum_{k=0}^{\infty}x^{2^k})^n+(\sum_{k=0}^{\infty}x^{2^k})^n\equiv1\mod2$

Let $A:=\sum_{k=0}^{\infty}x^{2^k}$. For what $n$ is it true that $(A+1)^n+A^n\equiv1\mod2$ (here we are basically working in $\mathbb{F}_2$.) The answer is all powers of 2, and it's fairly simple ...
4
votes
0answers
78 views

Pairwise sums are equal

The distinct positive integers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ with $n\ge2$ have the property that the $\binom{n}2$ sums $a_i+a_j$ are the same as the $\binom{n}2$ sums $b_i+b_j$ (in some order). ...
7
votes
1answer
662 views

Bernoulli numbers generating function

Consider the following generating formula: $$\frac{t}{e^t-1}=\sum_{n=1}^{\infty} B_n \frac{t^n}{n!}$$ There is some intuitive explanation about it? I want to know because I need to proof to myself ...
2
votes
0answers
58 views

Equation involved in generating function of divisor function [duplicate]

There is an identity between the divisor function of the odd numbers and the "odd" divisor function of power $3$(I don't know if there is a name for function for this type, if there is , sorry for my ...
2
votes
2answers
99 views

Generating function of a counting function.

Let $m$ be odd. Let $\eta(m)$ count the number of ways we can express $m$ as a product of exactly two odd numbers, counting order. What is $$\sum_{m\text{ odd }}\eta(m)x^m\text{ ? }$$ So, as an ...
6
votes
3answers
223 views

The ordinary generating function for $ζ(s)$

$$\zeta(s)^m = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$$ where $ζ(s)$ is the Riemann zeta function has the ordinary generating function: $$\sum \limits_{n=1}^{\infty} a_nx^n = x + {m \choose 1}\sum ...
1
vote
0answers
76 views

Is there a generic approach to Generating Function of periodic sequences?

Recently I read on wiki (see here): "Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones." ...
3
votes
5answers
714 views

How to create a generating function / closed form from this recurrence?

Let $f_n$ = $f_{n-1} + n + 6$ where $f_0 = 0$. I know $f_n = \frac{n^2+13n}{2}$ but I want to pretend I don't know this. How do I correctly turn this into a generating function / derive the closed ...
4
votes
1answer
203 views

Generating function for the divisor function

Earlier today on MathWorld (see eq. 17) I ran across the following expression, which gives a generating function for the divisor function $\sigma_k(n)$: $$\sum_{n=1}^{\infty} \sigma_k (n) x^n = ...
3
votes
2answers
617 views

Keep getting generating function wrong (making change for a dollar) [duplicate]

Possible Duplicate: Making Change for a Dollar (and other number partitioning problems) I am working on the classic coin problem where I would like to calculate the number of ways to make ...
6
votes
2answers
234 views

Partition of the natural numbers to arithmetic sequences

Let $\mathcal{S}$ be a collection of infinite arithmetic progression that forms a partition of $\mathbb{N}$. If $|\mathcal{S}|<\infty$ then $$\sum_{s\in\mathcal{S}:~d~\mbox{ is the difference of ...
2
votes
0answers
90 views

Proving generating functions equality

What do you use to prove the following equality (and possibly more general ones of the kind)? \begin{align*}\sum_{r,s,t} \frac{q^{r^2+rs+s^2+st+t^2}}{(q)_r (q)_s (q)_t} z_1^{r+s} z_2^{s+t} = ...
2
votes
3answers
1k views

The number of ways to write a positive integer as the sum of distinct parts with a fixed length

I am a topologist and not terribly familiar with the combo literature so please forgive me if this is standard. I'm hoping for some sort of reference for this. Given a positive integer $n$, I wish ...
4
votes
2answers
297 views

How to find generating function of cumulative sum?

Given the recurrence, $$ a_n = \begin{cases} 1, & n = 0 \\ a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n - 1)a_1 + na_0, & n \geq 1 \end{cases} $$ My attempt was, rewrite $a_n$ as: $$a_n = ...
4
votes
1answer
72 views

Generating Functions: how do I get my answers in terms of differential operators?

I'm reading and enjoying "generatingfunctionology". What a great fun book! But, I'm having some difficulty with the exercises. For example, take the series $a_n = n^2$ I'd like to find the Generating ...
5
votes
3answers
217 views

Two series relations, each one implies the other - from Andrews' partition book

That's my first question here, and i was encouraged to post because my question in MathOverflow (HERE) was beautifully and fast answered. But my questions in not at research level... As i said there, ...
5
votes
2answers
262 views

What's the name of this problem?

Problem: count the number of distinct ways to write number X as the sum of numbers {a, b, c...} with replacement. For instance, there are 3 ways to write 11: 2+2+7 2+2+2+2+3 2+3+3+3 And if I ...