Related papers: Hankel-type determinants for some combinatorial se…

Let $\{f_n\}$ be the Franel numbers given by $f_n=\sum_{k=0}^n\binom nk^3$, and let $p>5$ be a prime. In this paper we mainly determine $\sum_{k=0}^{p-1} \binom{2k}k\frac{f_k}{m^k}\pmod p$ for $m=5,-16,16,32,-49,50,96$. Let…

We consider the $n\times n$ Hankel matrix $H$ whose entries are defined by $H_{ij}=1/s_{i+j}$ where $s_k=(k-1)!$ and prove that $H$ is invertible for all $n\in\mathbb{N}$ by providing an explicit formula for its inverse matrix.

In recent preprints, Cigler considered certain Hankel determinants of convoluted Catalan numbers and conjectured identities for these determinants. In this note, we shall give a bijective proof of Cigler's Conjecture by interpreting…

For non-negative integers $r$ and $m$, let $S_m^{(r)}(n)$ denote the $r$-fold summation (or hyper-sum) over the first $n$ positive integers to the $m$th powers, with the initial condition $S_m^{(0)}(n) =n^m$. In this paper, we derive a new…

In this paper we introduce a class of determinants "of Hankel type". We use them to compute certain remarkable families of Drinfeld quasi-modular forms.

Let $p$ be a prime and $c,d\in\mathbb{Z}$. Sun introduced the determinant $D_p^-(c,d):=\det[(i^2+cij+dj^2)^{p-2}]_{1<i,j<p-1}$ for $p>3$. In this paper, we confirm three conjectures on $D_p^-(c,d)$ proposed by Zhi-Wei Sun.

In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and…

A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, where $F_1 = 1$, $F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$. For general recurrences…

A sequence $\{z_n\}_{n\geq0}$ is called ratio log-convex in the sense that the ratio sequence $\{\frac{z_{n+1}}{z_n}\}_{n\geq0}$ is log-convex. Based on a three-term recurrence for sequences, we develop techniques for dealing with the ratio…

The identity j/n {kn}\choose{n+j} =(k-1) {kn-1}\choose{n+j-1}- {kn-1}\choose{n+j} shows that j/n {kn}\choose{n+j} is always an integer. Here we give a combinatorial interpretation of this integer in terms of lattice paths, using a uniformly…

Let $\C{G}(z):=\sum_{n=0}^\infty z^{2^n}(1-z^{2^n})^{-1}$ denote the generating function of the ruler function, and $\C{F}(z):=\sum_{n=0}^\infty z^{2^n}(1+z^{2^n})^{-1}$; note that the special value $\C{F}(1/2)$ is the sum of the…

In this paper we resolve a conjecture of Zhi-Wei Sun concerning the integrality and arithmetic structure of certain trigonometric determinants. Our approach builds on techniques developed in our previous work, where trigonometric…

In this paper we study recurrences concerning the combinatorial sum $[n,r]_m=\sum_{k\equiv r (mod m)}\binom {n}{k}$ and the alternate sum $\sum_{k\equiv r (mod m)}(-1)^{(k-r)/m}\binom{n}{k}$, where m>0, $n\ge 0$ and r are integers. For…

In this paper, we study some supercongruences involving the sequence $$ t_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}2^k $$ and solve some open problems. For any odd prime $p$ and $p$-adic integer $x$, we determine…

Let $p>3$ be a prime and $(\frac{.}{p})$ be the Legendre symbol. For any integer $d$ with $p\nmid d$ and any positive integer $m$, Sun introduced the determinants…

In this paper, we confirm six conjectures on the exact values of some permanents, relating them to the Genocchi numbers of the first and second kinds as well as the Euler numbers. For example, we prove that…

We consider sums of the form \[\sum_{j=0}^{n-1}F_1(a_1n+b_1j+c_1)F_2(a_2n+b_2j+c_2)... F_k(a_kn+b_kj+c_k),\] in which each $\{F_i(n)\}$ is a sequence that satisfies a linear recurrence of degree $D(i)<\infty$, with constant coefficients. We…

In this paper we continue to investigate a certain class of Hankel-like positive definite kernels using their associated orthogonal polynomials. The main result of this paper is about the structure of this kind of kernels.

We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun and Z.-H. Sun. For instance, by using a transformation formula due to Chan and Zudilin, we show that for any…

In recent years, Z.-W. Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double…