Extended Pascal Triangles
Richard C. Bollinger · Mathematics Magazine · 1993
The Pascal triangle, with its associated properties, binomial coefficients, and Fibonacci numbers, is surely among the most familiar mathematical objects. Its versions, which arise in a fairly natural way, have entries and properties that are the natural generalizations of the original, and although not commonly known, are in many ways equally usefuil and interesting. In what follows we introduce these extended triangles and discuss a few of their properties and applications. The extended Pascal triangle, Tl,,, is the (left-justified) array of coefficients in the expansion of (1 + x + x2 + xx'-), for m n > O. These arrays may have been first explicitly discussed by J. E. Freund in a 1956 paper [1], where they arise in the solution of a restricted occupancy problem. There are no references to any previous similar development in that paper, and no connection with a generating function. N. Ya. Vilenkin's 1971 book on combinatorics [2, Chap. 5] discusses these generalizations (there called m-arithmetical triangles), in this case arriving at them through chessboard problems; again, they'are not connected with a generating function, and there are no references to previous work. S. J. Turner introduced these triangles (there called Pascal-T triangles, a name now also frequently used) in a 1979 paper [3] on a probability problem. In 1984 [4] and 1986 [5] the author proved a number of theorems on the counting properties of these arrays and gave examples of their use in combinatorics and reliability theory. Some problems related to the kinds of results discussed in this paper recently appear (although without an explicit connection to these triangles) in a collection [6] of problems on combinatorics by I. Tomescu (see, e.g., such problems as 1.9, 1.10, 1.11, 1.19).