## New answers tagged generating-functions

2
votes

### Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$

In terms of the finite and compactly supported measure $\mu=\sum x_j^2 \delta_{x_j^2}$, we are given the moments $\int t^k\, d\mu(t)$, $k=0,1,2 \ldots$.
Moment problems with compactly supported ...

2
votes

### Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$

[edit: completed] Assuming $x_i\ge0$ with $ \sum_i x_i <\infty$, we have that $\phi(t):=\sum_i(e^{x_it}-1)=\sum_{k\ge1} \big(\sum_i x_i^k\big)t^k/k!$ is an entire function (we can expand the ...

8
votes

### Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$

If $\sum x_i^2$ is finite, the sum $f(z)=\sum \frac{x_i^2}{1-x_i^2z}$ is a meromorphic function on the complex plane, and we know its Taylor series at 0. Thus we know $f$, hence the poles of $f$, ...

1
vote

### Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$

Here is what happens without assumptions.
Functions $p_n:=\sum_i x_i^n$ represent power-sum symmetric polynomials. By Newton's identities, we have
$$E(t):=\sum_{k=0}^\infty e_k \,t^k = \exp\left(\sum_{...

11
votes

Accepted

### Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$

If you plot $\log f_n$ versus $n$, with $f_n=\sum_i x_i^{2n}$, then the asymptotic slope for large $n$ will give you the largest of the $x_i$; subtracting that contribution from $f_n$ and repeating ...

3
votes

### Weak compositions with no subcomposition adding to (more than) $j$

Here is a way to obtain the generating function
$$ y=\sum_{N\geq 0}\sum_{k\geq 0}\kappa(N,k,4,2)t^k x^N. $$
From this formula (or from the method itself) you can extract a
formula for $\kappa(N,k,4,2)$...

2
votes

### Counting $m\times n$ $\bigl({1\atop1}{1\atop0}\bigr)$-free $(0,1)$-matrices

Command Master has already answered the question nicely. For my own understanding, this answer works out the details of that answer in the $m=3$ case (excluding the linear algebra at the end). Here's ...

7
votes

Accepted

### Counting $m\times n$ $\bigl({1\atop1}{1\atop0}\bigr)$-free $(0,1)$-matrices

We can show that for every $m$ there is a polynomial $q_m$ of degree $\binom m2$ such that $G_{m, n} = (m+1)^n q_m(n)$. We will do this by constructing a matrix $A_m$ such that $G_{m, n} = \mathbf{1} ...

0
votes

### Product of three or more independent sub-Gaussian varibles

As indicated in the comment, the boundedness can be used to prove that the product is sub-Gaussian. Suppose $X_{1}, \ldots, X_{n}$ are bounded such that for $i \in [n]$ there exists a constant such ...

3
votes

Accepted

### Ask for a generating function or an explicit expression of a triangle of positive integers

The generating function:
$${\cal C}(x,y) = \sum_{n,k\geq 0} C_{n,k} x^n y^{2k}$$
has the following explicit form:
$${\cal C}(x,y) = \frac{\arctan(y)}{y(1-x(1+y^2))}.$$
For "one more problem",...

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