Pascal三角形中的奇三角形

模p对称Pascal矩阵

Symmetric Pascal matrices modulo p

研究了有限域上元素为${i+j\choose i}$的二项式系数对称矩阵的特征多项式.

Pascal 三角形中的 Pascal 平铺与模 N 同余

Pascal tiling and congruences modulo N in Pascal's triangle

本文研究 Pascal 三角形前 N 行经几何变换所得矩阵的同余性质，证明其构成 Pascal 平铺当且仅当 N 为素数，并由此给出 Lucas 数同余式的替代证明，同时探讨 Fibonacci 伪素数情形及对多项式系数的推广。

列对齐杨辉三角形和Fibonacci序列的矩阵幂

Matrix Powers of Column-Justified Pascal Triangles and Fibonacci Sequences

如果L和R分别是元素为binom{i-1，j-1}和binom{i-1，n-j}的矩阵，则已知L^2 = I（mod 2），R^3 = I（mod 2），其中I是n维> 1的单位矩阵（参见P10735- 1999年5月的《美国数学月刊》）。我们将它推广到任何素数p，并给出了一个美丽的连接到斐波那契数。

与杨辉三角形有关的矩阵行列式的计算

Evaluations of some determinants of matrices related to the Pascal triangle

我们证明了几个评估的矩阵，其条目是由递归$a_{i,j}=a_{i-1,j}+a_{i,j-1}$，或其变化的行列式。这些评估由Roland Bacher进行了验证或扩展验证[J. Théorie Nombres Bordeaux 13（2001），to appear; see http：//arxiv.org/abs/math.CO/0109013]。

关于 $\mathbb{Z}$ 上元素源自二项式系数及 Catalan 数的矩阵的注记：分别考虑原始情形与模 $2$ 情形

A note on matrices over $\mathbb{Z}$ with entries stemming from binomial coefficients and from Catalan numbers once pure and once taken modulo $2$

本文在整数环中整理了 Pascal 矩阵及其模 $2$ 形式，以及由 Catalan 数定义的 Hankel 矩阵在原始与模 $2$ 情形下的共同特征，为研究仅含 $0$ 和 $1$ 元素的矩阵提供了动机。

与帕斯卡三角和斐波那契数列相关的三对角实对称矩阵

Tridiagonal real symmetric matrices with a connection to Pascal's triangle and the Fibonacci sequence

本研究推导了一类三对角实对称矩阵特征多项式的三项递推关系与闭式解，揭示其系数与帕斯卡三角对角元的关联，给出特征值包含关系的充分条件，并提出关联斐波那契数列的开放问题。

帕斯卡三角形行中最小公倍数的极值问题

Extremal Least Common Multiples in Rows of Pascal's Triangle

研究帕斯卡三角形行中选取n个项的最小公倍数的极值，给出渐近公式与结构描述，并验证了n=15和n=41处的有限结构转变。

典型色散与广义李雅普诺夫指数

Typical Dispersion and Generalized Lyapunov Exponents

设f（n）表示帕斯卡二项式三角形第n行中奇数元素的个数。我们研究f（n）的“平均色散”和“典型色散”--后者涉及计算广义李雅普诺夫指数--然后转向高维例子的数值分析。

Divisibility Properties of Integer Sequences

A sequence of nonzero integers $f = (f_1, f_2, \dots)$ is ``binomid'' if every $f$-binomid coefficient $\left[\! \begin{array}{c} n \\ k \end{array}\! \right]_f$ is an integer. Those terms are the generalized binomial coefficients: \[ \left[\! \begin{array}{c} n \\ k \end{array}\! \right]_f \ = \ \frac{ f_nf_{n-1}\cdots f_{n-k+1} }{ f_kf_{k-1}\cdots f_1 }. \] Let $Δ(f)$ be the infinite triangle with those numbers as entries. When $I = (1, 2, 3, \dots)$ then $Δ(I)$ is Pascal's Triangle so that $I$ is binomid. Surprisingly, every row and column of Pascal's Triangle is also binomid. For any $f$, …

A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence

The Stern polynomials defined by $s(0;x)=0$, $s(1;x)=1$, and for $n\geq 1$ by $s(2n;x)=s(n;x^2)$ and $s(2n+1;x)=x\,s(n;x^2)+s(n+1;x^2)$ have only 0 and 1 as coefficients. We construct an infinite lower-triangular matrix related to the coefficients of the $s(n;x)$ and show that its inverse has only 0, 1, and $-1$ as entries, which we find explicitly. In particular, the sign distribution of the entries is determined by the Prouhet-Thue-Morse sequence. We also obtain other properties of this matrix and a related Pascal-type matrix that involve the Catalan, Stirling, Fibonacci, Fine, and Padovan n…

Divisibility of binomial coefficients by powers of two

For nonnegative integers $j$ and $n$ let $Θ(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are not divisible by $2^{j+1}$. In this paper we prove that the family $j\mapstoΘ(j,n)$ usually follows a normal distribution. The method used for proving this theorem involves the computation of first and second moments of $Θ(j,n)$, and uses asymptotic analysis of multivariate generating functions by complex analytic methods, building on earlier work by Drmota (1994) and Drmota, Kauers and Spiegelhofer (2016).