证明C(0,n)+C(1,n+1)+C(2,n+2)+...+C(k,n+k)=C(k,n+k+1)
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C(k,n+k+1) = C(k-1,n+k) + C(k,n+k)
= C(k,n+k) + [C(k-1,n+k-1) + C(k-2,n+k-1)]
= C(k,n+k) + C(k-1,n+k-1) + [C(k-2,n+k-2) + C(k-3,n+k-2)]
= C(k,n+k) + C(k-1,n+k-1) + C(k-2,n+k-2) + .+ C(1,n+1) + C(0,n+1)
= C(k,n+k) + C(k-1,n+k-1) + C(k-2,n+k-2) + .+ C(1,n+1) + C(0,n)
= C(k,n+k) + [C(k-1,n+k-1) + C(k-2,n+k-1)]
= C(k,n+k) + C(k-1,n+k-1) + [C(k-2,n+k-2) + C(k-3,n+k-2)]
= C(k,n+k) + C(k-1,n+k-1) + C(k-2,n+k-2) + .+ C(1,n+1) + C(0,n+1)
= C(k,n+k) + C(k-1,n+k-1) + C(k-2,n+k-2) + .+ C(1,n+1) + C(0,n)
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