SF13

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Evaluate S=\sum _{{k=0}}^{n}{n \choose k}^{2}\,

Identity: {n \choose k}={\frac  {(-1)^{k}(-n)_{k}}{k!}}\,

S=\sum _{{k=0}}^{n}{\frac  {(-n)_{k}(-n)_{k}}{k!k!}}=\sum _{{k=0}}^{n}{\frac  {(-n)_{k}(-n)_{k}}{(1)_{k}}}{\frac  {1^{k}}{k!}}\,

S={}_{2}F_{1}(-n,-n;1;1)\, because the terms in the sum of the generalized hypergeometric function equal 0 when k>n\,

Since {}_{2}F_{1}(a,b;c;1)={\frac  {\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}}\,

S={\frac  {\Gamma (1)\Gamma (2n+1)}{\Gamma (n+1)\Gamma (n+1)}}={\frac  {(2n)!}{n!n!}}={\frac  {2^{{2n}}\left({\frac  {1}{2}}\right)_{n}}{n!}}\,

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