# Tagged Questions

**3**

votes

**1**answer

243 views

### Is there a nice way to write the generating function obtained by taking the quadratic coefficients of another one?

Suppose that you have a generating function
$$
f(q) = \sum_{k=0}^\infty a_k q^k
$$
It's not too hard to obtain the generating function
$$
f_{n,m}(q) = \sum_{k=0}^\infty a_{nk + m}q^k
$$
by taking a ...

**1**

vote

**1**answer

238 views

### An infinite product: combinatorial interpretation

It is an undergraduate exercise to show that the generating function for the sequence of unrestricted integer partitions $p(n)$ is the celebrated infinite product
...

**2**

votes

**2**answers

826 views

### Number of 1 in binary representation of n

Let $1(n)$ be the number of digits $1$ in binary representation of number $n$.
For example, $13=1101_2$ so $1(13)=3\\$
Is there explicit form of $\,\,\sum{1(i)x^i} $?
I checked OEIS and didn't find ...

**3**

votes

**0**answers

190 views

### Generating function for the characteristic function of prime numbers

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:
...

**6**

votes

**1**answer

236 views

### Does the following operation on modular forms yield something modular?

Let $f(z) = \sum_{n=0}^\infty a_nq^n$ be the fourier expansion of a (quasi-)modular form (with $q = e^{2\pi i z}$). Consider the following related functions:
$$f_{m,k}(z) = \sum_{n=0}^\infty a_{mn + ...

**0**

votes

**1**answer

145 views

### Given a generating function with “zeros”, can one derive the function for ONLY the “zeros”?

If I have an generating function (GF) --- ordinary or exponential --- defining a series with at least one coefficient equal to zero, is there a general method to find the "inverse GF", i.e., the GF ...

**8**

votes

**4**answers

458 views

### Partitions-Sum of divisors identity

A few years ago I first read about the marvelous Euler identity:
$\sum_{n\in\mathbb{N}}p(n)z^n=\prod_{k\geq1}\frac{1}{1-z^k}$,
where $p(n)$ is the number of partitions of $n$ ($p(0)=1$ by ...

**2**

votes

**2**answers

757 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} = ...

**4**

votes

**0**answers

224 views

### Generating function related to 2-residues of partitions

Question
Express the following power series in two commuting variables $x,y$ as an infinite product, or a short sum of infinite products:
$$
...

**5**

votes

**2**answers

824 views

### What are the best known bounds on the number of partitions of $n$ into exactly $k$ distinct parts?

For example, if $n = 10$ and $k = 3$, then the legal partitions are
$$10 = 7 + 2 + 1 = 6 + 3 + 1 = 5 + 4 + 1 = 5 + 3 + 2$$
so the answer is $4$. By choosing $k$ random elements of $\{1,\ldots,2n/k\}$, ...

**8**

votes

**3**answers

563 views

### Identifying the generating function $ G(a,z) = \sum_{n=0}^{\infty} a^n z^{(n+1)(n+2)/2}. $

I have computed a generating function for a problem involving a particular series, and would like to know if anyone has any references or a categorisation for it? It's
$$
G(a,z) = \sum_{n=0}^{\infty} ...

**2**

votes

**0**answers

219 views

### Algebraic Dirichlet series and beyond

I wonder what the "right" notion of "algebraic Dirichlet series" might be. Here I'm thinking of formal Dirichlet series $D(s)=\sum_{n\geq 1} a_n/n^s$, say with $a_n$ being rational numbers.
I'm ...

**13**

votes

**1**answer

832 views

### How many pairs (M, N) of sets of size n have M + N = {0, 1, …, n^2-1}?

How many pairs (M, N) of sets of size n have M + N = {0, 1, ..., n^2-1}?
Manfred Schroeder, in Number Theory in Science and Communication, 4th edition, asks (p. 27): find all pairs of sets $(M,N)$, ...