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Accepted

• 11.6k

• 52.6k
Accepted

Hankel determinant evaluation of special lattice paths

The $d(n):=D(n,n)$ is OEIS sequence A005773. The Hankel determinant property is given in the sequence entry. Also the recursion $\;nd_{n}=2nd_{n-1}+3(n-2)d_{n-2}$. A proof could come from a similar ...
• 2,359

Some curious Hankel determinants

The identity is clearly invariant to scaling $a$ by a constant. (One has to check the constant appears in the determinant to power $n(n+1)$). So wlog we may assume $a(1)=1$. We can expand $\log a$ ...
• 126k
Accepted

• 40.2k

Determinant of a checkerboard Hankel matrix with Catalan numbers

If you consider the operator on functions that sends $f(x)$ to $$Tf(y) = \int_{-2}^2 \frac{ \sqrt{4-x^2} }{2\pi } \frac{1}{ 1- \alpha x y} f(\alpha x) dx$$ then this is a well-defined integral ...
• 126k

Hankel determinants of harmonic numbers

As asked by Fedor Petrov I sketch the missing details. If $a(n)$ is any sequence with $a(0)=1$, such that all Hankel determinants $M_n=\det\left(a(i+j)\right)_{i,j=0}^n$ are $\neq 0$, define a ...
• 5,440
Accepted

Some nice polynomials related to Hankel determinants

See Example 4 in Section 3.1.6 of https://arxiv.org/abs/1409.2562 (Federico Ardila, "Algebraic and geometric methods in enumerative combinatorics"). It explains that your $g_n(k)$ has an ...
• 20.7k

A continued J fraction for $a_n = \frac{1}{(n+1)^2}$?

Sagemath can do that too ...
• 3,447

• 2,359

Hankel determinants of binomial coefficients

This is not an answer. Although there might be no "closed formula" for the case $p=4$, I wish to add the following to the case $p=2$ for which the determinants evaluate to $2^{n-1}$. Suppose we ...
• 40.2k

Moment problem on [-1,1]: necessary and sufficient conditions

The following paper answers a more general version of this question, namely the Hausdorff moment problem for compact sets in $\mathbb{R}^n$. This is a fairly broad topic, and you may enjoy learning ...
• 28.1k

Some Hankel Determinants

Not yet a fully satisfying answer, but Theorem 1 does generalize. Given two sequences $j$ and $k$, define the sequence $h$ by $$h(m)=j(m-1)k(m)$$ Let $H$, $J$ and $K$ be the corresponding families ...
• 21.5k