Tea Tasting and Pascal Triangles

Tea Tasting and Pascal Triangles

The Pascal-de Moivre Triangles*

The coefficients of the Pascal triangle were generalized in 1756 by de Moivre [5]. Each row of a Pascal triangle contains a sequence of numbers that are the coefficients of the power series expansion for the binary expression (l + x)^. The de Moivre formula [2], [4], [5], [6] derives the coefficients of the power series for the generalized expansion of (1 + x + x + • • • + x^"^). Thus, for inte...

Generalized Pascal triangles and Toeplitz matrices

The purpose of this article is to study determinants of matrices which are known as generalized Pascal triangles (see R. Bacher. Determinants of matrices related to the Pascal triangle. J. Théor. Nombres Bordeaux, 14:19–41, 2002). This article presents a factorization by expressing such a matrix as a product of a unipotent lower triangular matrix, a Toeplitz matrix, and a unipotent upper triang...

Tasting Tea By SIR RONALD

Large primes in generalized Pascal triangles

In this paper, after presenting the results of the generalization of Pascal triangle (using powers of base numbers), we examine some properties of the 112-based triangle, most of all regarding to prime numbers. Additionally, an effective implementation of ECPP method is presented which enables Magma computer algebra system to prove the primality of numbers with more than 1000 decimal digits. 1 ...