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Give a combinatorial proof & proof using generating functions of the following identity:

$$\sum_{k=1}^n k {n\choose k}^2 = n {2n-1 \choose n-1}.$$

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Is this homework? –  Phira Apr 23 '11 at 20:56
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Do you know what the terms mean? Can you give us some background? Do you know the corresponding proofs of the Vandermonde identity? –  Phira Apr 23 '11 at 20:59
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Well, if you don't know whether you know what $C(n,k)$ means or what the summation index of your sum is, or whether you know the corresponding proof of the Vandermonde identity, I don't know why you are asking the question. Do you think that I will explain your question to you? –  Phira Apr 23 '11 at 21:15
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As you alluded, it looks like the Vandermonde identity. Look at the proofs (combinatorial and generating function respectively) of that identity and see how you can deal with the $k$. –  Mitch Apr 23 '11 at 21:26
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Please do not post questions in the imperative mode. You are not assigning us homework. Rather than "give a proof..." say "how do I prove...." –  Qiaochu Yuan Apr 23 '11 at 22:37
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closed as not a real question by Andres Caicedo, t.b., Eric Naslund, Zev Chonoles, J. M. Apr 26 '11 at 12:54

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1 Answer

Combinatorial proof:

Adapt the combinatorial proof for the Vandermonde identity by kissing one of the chosen men.

Proof by generating functions:

Adapt the proof for the Vandermonde identity by differentiating both sides.

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Kissing? Men? Not president of the committee? –  Mitch Apr 24 '11 at 1:25
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