推 ooopppeeennn:太感恩了~~~~~~~~~ 04/12 14:25
→ ooopppeeennn:17994那一題可以順便嗎?? 04/12 14:25
※ 引述《ooopppeeennn (open)》之銘言：
: 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 兩邊同對x微分
(-1)* (1-x)^(-2) *(-1) = (1-x)^(-2) = sum k=0to∞ k * x^ (k-1) 再同對x微分
(-2)* (1-x)^(-3) *(-1) = 2(1-x)^(-3) = sum k=0to∞ k(k-1) * x^ (k-2)
兩邊同乘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.同對x微 2.同乘x 3同對x微 4.同乘x
: 3. 0,1*3,2*4,...,k*(k+2)
1.同對x微 2.同乘x^3 3.同對x微 4.同除x^2
: 4. 0, 1*2*3,2*3*4,...,k*(k+1)*(k+2)
1.同對x微 2.同乘x^2 3.同對x微 4.同乘x^2 5.同對x微 6.同除x
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