论文标题
帕斯卡尔（Pascal
Binomial Coefficients in a Row of Pascal's Triangle from Extension of Power of Eleven: Newton's Unfinished Work
论文作者
论文摘要
本文的目的是找到一个通用公式来生成任何一排Pascal的三角形，以作为$ \ left（11 \ right）^{n} $的概念的扩展。在这项研究中，通过将11的功率的概念扩展到101、1001、10001等的功率，从而显示了每行Pascal的三角形的可视化。我们简要讨论我们提出的概念如何通过插入$ 1 $ 1 $至1 $（十一）之间的适当数量的零来适用于任何$ n $，这就是$ \ left（11 \ right）^{n} $的概念，已扩展到$ \ left（1 \ theta1 \ theta1 \ right）^n} $，其中$θ$是$θ$的代表。我们提出了一个用于获得$θ$的公式。拟议的概念已通过帕斯卡尔（Pascal）的三角形进行了验证，并成功匹配。最后，考虑到$ 51^{\ text {st}} $ row作为示例，帕斯卡尔（Pascal）的三角形的三角形。
The aim of this paper is to find a general formula to generate any row of Pascal's triangle as an extension of the concept of $\left(11\right)^{n}$. In this study, the visualization of each row of Pascal's triangle has been presented by extending the concept of the power of 11 to the power of 101, 1001, 10001, and so on. We briefly discuss how our proposed concept works for any $n$ by inserting an appropriate number of zeros between $1$ and $1$ (eleven), that is the concept of $\left(11\right)^{n}$ has been extended to $\left(1\Theta1\right)^{n}$, where $Θ$ represents the number of zeros. We have proposed a formula for obtaining the value of $Θ$. The proposed concept has been verified with Pascal's triangle and matched successfully. Finally, Pascal's triangle for a large n has been presented considering the $51^{\text{st}}$ row as an example.
