Extended Pascal Triangles

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