※ 引述《ooopppeeennn (open)》之铭言： : Find the ordinary generating functions of the sequence : 1. 0,0,2*1,3*2,...,k*(k-1) 即求 sum k=0to∞ k(k-1) * x^k (1-x)^(-1) = sum k=0to∞ x^k 两边同对x微分 (-1)* (1-x)^(-2) *(-1) = (1-x)^(-2) = sum k=0to∞ k * x^ (k-1) 再同对x微分 (-2)* (1-x)^(-3) *(-1) = 2(1-x)^(-3) = sum k=0to∞ k(k-1) * x^ (k-2) 两边同乘x^2 => 2x(1-x)^(-3) = sum k=0to∞ k(k-1) * x^k : 2. 0,1^2,2^2,3^2,...,k^2 1.同对x微 2.同乘x 3同对x微 4.同乘x : 3. 0,1*3,2*4,...,k*(k+2) 1.同对x微 2.同乘x^3 3.同对x微 4.同除x^2 : 4. 0, 1*2*3,2*3*4,...,k*(k+1)*(k+2) 1.同对x微 2.同乘x^2 3.同对x微 4.同乘x^2 5.同对x微 6.同除x --

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