Iterated Rascal triangles (Q6590675)

In mathematics, the Pascal triangle is an infinite triangular array of binomial coefficients that plays a crucial role in probability theory, combinatorics, and algebra. Pascal triangles also arise in many applications such as binomial expansion, probability, combinatorics, linear algebra, electrical engineering, and order statistics. The Pascal triangle determines the coefficients which arise in binomial expansions. A second useful application of the Pascal triangle is in the calculation of combinations. The Pascal triangle has many properties and contains many patterns of numbers.\N\NThe Rascal triangle in Eq. (1.2), which is a modified version of the Pascal triangle, possesses a multitude of mathematical properties. One example is that the third column(s) from the center is always a perfect square. The rows of the Rascal triangle have interior entries which are determined by a so-called diamond formula in Eq. (1.1). There are many more examples of Rascal identities, some of which have appeared in other areas of mathematics and the Rascal triangle itself is a mathematical treasure trove of exploration and inquiry-based learning.\N\NIn the related paper, the authors first present a natural extension of the Rascal triangle that directly connects the Rascal triangle to the Pascal triangle. Then, they establish an infinite sequence of number triangles \( \{R_{i}\}_{i=0}^{\infty }\) in which \(R_{1}\) is the Rascal triangle and the limit of the sequence is a Pascal triangle, which are called iterated Rascal triangles. Also, they provide an unexpected and elegant recursive formula (Theorem 2.1) that generates the entries in any iterated Rascal triangle. Moreover, the authors derive some formulas, generating functions, and combinatorial identities for them. In addition, a version of Lucas' theorem (Theorem 5.1) for the iterated Rascal triangles is proved. Furthermore, the authors consider an identity from the Rascal triangle that has a well-known analogous identity in the Pascal triangle and extend it to all iterated Rascal triangles.\N\NEven though it is possible to make many, the paper is concluded with one conjecture: Generalizations of these \(R_{1}\) identities to all \(R_{i}\) and even \(R_{i,(c,d\ast ,d_{1},d_{2})}\) remain open. The Rascal triangle is both a quintessential example and a wellspring of inquiry-based learning. What other Rascal-Pascal identities can be discovered?