推 lio00567 :感謝!! 10/23 01:23
※ 引述《lio00567 (隨便)》之銘言：
: Give a combinational proof that if n is a positive integer,then
: n
: Σ k^2(n,K)=n(n+1)2^(n-2)
: k=1
: 想了好久都想不出來 麻煩各位大大解惑了!!
(1+x)^n = Σ[k;0,n] x^k C(n,k)
differentiate it once
n(1+x)^{n-1} = Σ[k;1,n] k x^{k-1} C(n,k)
2n*2^{n-2} = Σ[k;1,n] k C(n,k)
differentiate it twice
n(n-1)(1+x)^{n-2} = Σ[k;2,n] k(k-1) x^{k-2} C(n,k)
n(n-1)2^{n-2} = Σ[k;2,n] k(k-1) C(n,k)
hence,
Σ[k;1,n] k^2 C(n,k) = n(n+1) 2^{n-2}
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