Ansatz in a Nutshell: A comprehensive step-by-step guide to polynomial, $C$-finite, holonomic, and $C^2$-finite sequences

Sequences: Polynomial, C-finite, Holonomic, ...

Polynomial, C-finite, Holonomic are the most common ansatz to describe the pattern of the sequences. We propose a new ansatz called X-recursive that generalize those we mentioned. We also discuss its closure properties and compare this ansatz to another new similar ansatz from another paper.

C-有限算子满足完整算子

The C-finite Ansatz Meets the Holonomic Ansatz

我们展示了如何连续的Almkvist-Zeilberger算法可以用来有效地发现和证明满足生成函数的序列定义为C-有限多项式序列（如切比雪夫多项式）的幂的积分的微分方程，可以自动推导出线性递归多项式系数的序列本身。我们用Maple包CfiniteIntergal.txt完全实现了这一点。

拟完整解与限制格点游动

The Quasi-Holonomic Ansatz and Restricted Lattice Walks

伟大的计数器学家热尔曼·克雷韦拉斯（Germain Kreweras）凭经验发现了这一有趣的事实，然后需要大量的篇幅和人类的聪明才智来证明这一点。其他伟大的计数器学家，例如海因里希·尼德豪森（Heinrich Niederhausen）、艾拉·杰塞尔（Ira Gessel）和米雷耶·布斯克-梅卢（Mireille Bousquet-Mélou），找到了其他巧妙的、“更简单”的证明。但没有一个像我们这样简单！我们的证明（在我们忠实的计算机的慷慨帮助下）在传统意义上是“丑陋的”，因为对于一个卑微的人来说，遵循所有的步骤是痛苦的。但根据我们卑微的审美品味，这个证明要优雅得多，因为它（概念上）是单行的。因此，如果这条线相当长（一个由一般计数函数满足的巨大的部分递归方程），它占用的存储空间比一张分辨率很低的照片少。

Zeilberger的Pfweenian完整定位器

Zeilberger's Holonomic Ansatz for Pfaffians

一个变化的Zeilberger的完整的符号行列式的评估提出了专门处理Pfiderans。该方法也适用于反对称矩阵的行列式，原来的方法不工作。由于Zeilberger的方法是基于行列式的拉普拉斯展开式（余因子展开），我们从Pfidian的余因子展开得到我们的方法。为了证明我们的方法的力量，我们证明，使用计算机代数算法，在论文中提出的一些结构“Pfiran分解和Pfiran模拟的q-Catalan汉克尔行列式”石川，田川，曾。一个与划分和Motzkin路径相关的小求和公式作为推论如下。

Binomial Determinants for Tiling Problems Yield to the Holonomic Ansatz

We present and prove closed form expressions for some families of binomial determinants with signed Kronecker deltas that are located along an arbitrary diagonal in the corresponding matrix. They count cyclically symmetric rhombus tilings of hexagonal regions with triangular holes. We extend a previous systematic study of these families, where the locations of the Kronecker deltas depended on an additional parameter, to families with negative Kronecker deltas. By adapting Zeilberger's holonomic ansatz to make it work for our problems, we can take full advantage of computer algebra tools for sy…

On the Average Maximal Number of Balls in a Bin Resulting from Throwing r Balls into n Bins T times

We use the holonomic ansatz to estimate the asymptotic behavior, in $T$, of the average maximal number of balls in a bin that is obtained when one throws uniformly at random (without replacement) $r$ balls into $n$ bins, $T$ times. Our approach works, in principle, for any fixed $n$ and $r$. We were able to do the cases $(n,r)$ = $(2,1),(3,1),(4,1), (4,2)$, but things get too complicated for larger values of $n$ and $r$. We are pledging a \$150 donation to the OEIS for an explicit expression, (in terms of $n$, $r$, and $π$) for the constant $C_{n,r}$ such that that average equals $\frac{n}{r}\…