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1×C(n,1)+3×C(n,2).+(2n-1)C(n,n)求和
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1×C(n,1)+3×C(n,2).+(2n-1)C(n,n)求和
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答案和解析
原式
=2(∑iC(n,i))-∑C(n,i) (i从1到n)
=2(∑nC(n-1,i-1))-(2^n - 1) (i从1到n)
=2n(∑C(n-1,j))-2^n + 1 (j从0到n-1)
=2n*2^(n-1)-2^n+1
=(n-1)*2^n+1
=2(∑iC(n,i))-∑C(n,i) (i从1到n)
=2(∑nC(n-1,i-1))-(2^n - 1) (i从1到n)
=2n(∑C(n-1,j))-2^n + 1 (j从0到n-1)
=2n*2^(n-1)-2^n+1
=(n-1)*2^n+1
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