Several authors are currently working on generalized Appell polynomials and their applications in the framework of hypercomplex function theory in Rⁿ⁺¹. A few years ago, two of the authors of this paper introduced a prototype of these generalized Appell polynomials, which heavily draws on a one-parameter family of non-symmetric number triangles 𝒯(n), n ≥ 2. In this paper, we prove several new and interesting properties of finite and infinite sums constructed from entries of 𝒯(n), similar to the ordinary Pascal triangle, which is not a part of that family. In particular, we obtain a recurrence relation for a family of finite sums, analogous to the ordinary Fibonacci sequence, and derive its corresponding generating function.