[Combinatorics] Jia Xian's Triangle
By: Tao Steven Zheng (鄭濤)
【Problem】
The arithmetic triangle is a diagram that illustrates the binomial coefficients of the expansion of . Jia Xian (fl. 1050 AD) is known to have discussed the arithmetic triangle in his works. The combinatorial formula for calculating the binomial coefficients in the arithmetic triangle is
where is the row and
is the column. Notice that by adding two adjacent coefficients in a row results in the coefficient below the two:
Prove this fact.
【Solution】
Therefore,
【Historical Note】
The Arithmetic Triangle has many names. In China, it is called Jia Xian's Triangle or Yang Hui's Triangle, named after the mathematicians Jia Xian (c. 1010 - 1070 AD), who first discovered it, and Yang Hui (1238 - 1298 AD), who commented on Jia Xian's lost works. In Iran, it is called the Khayyam's Triangle, named after the Persian astronomer and poet Omar Khayyam (1048 - 1131 AD). In Europe, it is called the Pascal's Triangle, named after the French mathematician, inventor, and philosopher Blaise Pascal (1623 - 1662 AD).
