An Investigation of Sequences Derived from Hoggatt Sums and Hoggatt Triangles

In a recent note [1], the authors discuss derivations of integer sequences called Hoggatt Sums and associated triangular arrays called Hoggatt Triangles. The nomenclature was proposed as a tribute to the late Verner Hoggatt, Jr. since the investigation and extension of an unpublicized conjecture of Hoggatt ultimately resulted in the above sums and triangles. In personal correspondence [2], Hoggatt conjectured that the third (counting as 0, 1, 2, 3, …) right diagonal of Pascal’s triangle could be used to determine the sequence of integers, S0, S1, S2, … S m ,…, which are identically the Baxter permutation counts [3] of indices 0, 1, 2, …, m, … Hoggatt based his calculation algorithm for S m on sums of products between third diagonal terms from Pascal’s triangle and appropriately corresponding terms from a completed S m -1. The authors’ note [1] supplied the missing proof of Hoggatt’s conjecture. Hoggatt’s conjecture was then extended to include all right Pascal triangle diagonals indexed as 0, 1, 2, 3, …, d, …. For each d, the set of S m ’s became Hoggatt sums of order d, and the individual integers which sum to a particular S m became row members of a triangular array called a Hoggatt triangle of order d. With the inclusion of d as a variable parameter, the numerical results of [1] can be interpreted as sequences of (S d ) m ’s with Fixed index d and variable index m. For example, the Baxter permutation count values are Hoggatt sums of order three whose general sequence term is (S3) m . Sequences of Hoggatt sums follow a linear recursion which is index-variant in m, i.e., the calculation of (S d ) m for d fixed depends not only on previous members of the sequence but also depends on the value of m. Difference equations for this type of recursion are known to be difficult, if not impossible, to obtain by operational methods [4].