利用切比雪夫多项式证明某些类benzenches的Kekulé数
Proving some conjectures on Kekulé numbers for certain benzenoids by using Chebyshev polynomials
摘要：
在化学中，Cyvin-Gutman列举了某些苯并芘的凯库勒数，并在OEIS上将其记录为$A050446$。这个数字正好是递归$T(n, m) = T(n, m-1) + \sum^{\lfloor\frac{n-1}{2}\rfloor}_{k=0} T(2k, m-1)T(n-1-2k, m)$定义的双变量数组$T(n,m)$，其中$T(n,0)=T(0,m)=1$表示所有非负整数$m,n$。有趣的是，这个数字也出现在加权图，图多面体，魔术标号和单位本原矩阵的背景下，由不同的作者研究。在OEIS上做了几个有趣的演示。这些结构与$T(n,m)$的行和列母函数有关。本文给出了列母函数的显式表达式，它也是Bóna，Ju和Yoshida研究的母函数$F(n,x)$.利用第二类切比雪夫多项式，我们也得到了非线性函数的表示.这使我们能够证明所有这些假设。
英文摘要：
In chemistry, Cyvin-Gutman enumerates Kekulé numbers for certain benzenoids and record it as $A050446$ on OEIS. This number is exactly the two variable array $T(n,m)$ defined by the recursion $T(n, m) = T(n, m-1) + \sum^{\lfloor\frac{n-1}{2}\rfloor}_{k=0} T(2k, m-1)T(n-1-2k, m)$, where $T(n,0)=T(0,m)=1$ for all nonnegative integers $m,n$. Interestingly, this number also appeared in the context of weighted graphs, graph polytopes, magic labellings, and unit primitive matrices, studied by different authors. Several interesting conjectures were made on the OEIS. These conjectures are related to both the row and column generating function of $T(n,m)$. In this paper, give explicit formula of the column generating function, which is also the generating function $F(n,x)$ studied by Bóna, Ju, and Yoshida. We also get trig function representations by using Chebyshev polynomials of the second kind. This allows us to prove all these conjectures.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
