Figuring more prominently in antiquity than in modern mathematics—though there are exceptions from Gauss to Euler to Cauchy, et al.—the subject of polygonal numbers is one that has long intrigued me. A subset of the more general figurate numbers, they consist of integer sequences defined by numbers that can be represented as geometric figures, regular polygons in this case. For example, the triangular numbers are those for which a number of pebbles can be arranged in the shape of a triangle; the squares, represented as squares; the pentagonal as a pentagon; and so on for the rest.
Nicomachus noted that for a k-sided polygon, the nth polygonal number is formed by taking the sum of every (k-2)th number from 1 to n. That is, for the triangular numbers, k=3, thus they arise from summing every number from 1 to n: 3 (1+2), 6 (1+2+3), 10 (1+2+3+4), 15 (1+2+3+4+5), etc. For the squares, k=4, you take the sum of every second number, thus the sum of odd numbers: 4 (1+3), 9 (1+3+5), 16 (1+3+5+7), 25 (1+3+5+7+9). For any k, the nth number to add to the sum is (k-2)n-(k-3), so we can define:
The traditional factorial function, designated n!, gives the product of the natural or counting numbers from 1 to n. Out of curiosity, I decided to explore a generalization of this function where we’re taking the product of polygonal numbers from 1 to n for some k; for p(n,2) we’ll still yield the traditional factorial, but for k>2, we will produce new integer sequences of the product of the triangular, square, pentagonal, hexagonal, etc., numbers. With a bit of exploration and work, I discovered that these can be generated via the closed form:
where (m)n represents the Pochhammer symbol.
A few of these sequences have already been studied in other contexts—e.g., the product of the triangulars is sequence A006472; the squares are sequence A001044; the hexagonals are sequence A000680, etc.—but most are novel. Of perhaps more or equal interest are the interrelationships discovered between some of these sequences. The ratio of the hexagorials to the trigorials, for instance, produces the Catalan numbers (proof). The infinite sum of the ratio of the tetragorials to the hexagorials can yield π: