パスカルの三角形
( x + y ) n + 1 = x ( x + y ) n + y ( x + y ) n = ∑ i = 0 n a i x n − i + 1 y i + ∑ i = 0 n a i x n − i y i + 1 = ∑ i = 0 n a i x n − i + 1 y i + ∑ i = 1 n + 1 a i − 1 x n − i + 1 y i = a 0 x n + 1 + ∑ i = 1 n a i x n − i + 1 y i + ∑ i = 1 n a i − 1 x n − i + 1 y i + a n y n + 1 = x n + 1 + ∑ i = 1 n ( a i − 1 + a i ) x n − i + 1 y i + y n + 1 {\displaystyle {\begin{aligned}(x+y)^{n+1}&=x(x+y)^{n}+y(x+y)^{n}\\&=\sum _{i=0}^{n}a_{i}x^{n-i+1}y^{i}+\sum _{i=0}^{n}a_{i}x^{n-i}y^{i+1}\\&=\sum _{i=0}^{n}a_{i}x^{n-i+1}y^{i}+\sum _{i=1}^{n+1}a_{i-1}x^{n-i+1}y^{i}\\&=a_{0}x^{n+1}+\sum _{i=1}^{n}a_{i}x^{n-i+1}y^{i}+\sum _{i=1}^{n}a_{i-1}x^{n-i+1}y^{i}+a_{n}y^{n+1}\\&=x^{n+1}+\sum _{i=1}^{n}(a_{i-1}+a_{i})x^{n-i+1}y^{i}+y^{n+1}\end{aligned}}}