Question Number 26133 by abdo imad last updated on 21/Dec/17
$${find}\:{the}\:{value}\:{of}\:\:\left({C}_{{n}} ^{\mathrm{0}\:\:} \right)^{\mathrm{2}} \:+\left({C}_{{n}} ^{\mathrm{1}} \right)^{\mathrm{2}} \:+\left({C}_{{n}} ^{\mathrm{2}} \right)^{\mathrm{2}} \:+...\left({C}_{{n}} ^{{n}} \right)^{\mathrm{2}} . \\ $$
Commented by abdo imad last updated on 21/Dec/17
$${let}\:{put}\:\:{A}_{{n}} =\left(\:{C}_{{n}} ^{\mathrm{0}} \:\right)^{\mathrm{2}} \:\:+\left(\:{C}_{{n}} ^{\mathrm{1}} \:\right)^{\mathrm{2}} \:\:+...+\left({C}_{{n}} ^{{n}} \:\right)^{\mathrm{2}} \\ $$$${we}\:{have}\:\:\left(\mathrm{1}+{x}\right)^{\mathrm{2}{n}} \:=\:\:\sum_{{k}=\mathrm{0}} ^{{k}=\mathrm{2}{n}} \:{x}^{{k}} \:\:=\:{C}_{\mathrm{2}{n}} ^{\mathrm{0}} +{C}_{\mathrm{2}{n}} ^{\mathrm{1}} {x}\:+{C}_{\mathrm{2}{n}} ^{\mathrm{2}} {x}^{\mathrm{2}} \:+...{C}_{\mathrm{2}{n}} ^{\mathrm{2}{n}} \:{x}^{\mathrm{2}{n}} \\ $$$${but}\:\left(\mathrm{1}+{x}\right)^{\mathrm{2}{n}} =\left(\mathrm{1}+{x}\right)^{{n}} .\left(\mathrm{1}+{x}\right)^{{n}} =\left(\:\sum_{{k}=\mathrm{0}} ^{{k}={n}} \:{C}_{{n}} ^{{k}} \:{x}^{{k}} \right)\left(\sum_{{k}=\mathrm{0}} ^{{k}={n}} \:\:{C}_{{n}} ^{{k}} \:{x}^{{k}} \right) \\ $$$$=\:\sum_{{k}=\mathrm{0}} ^{{k}=\mathrm{2}{n}} \:\lambda_{{k}} \:{x}^{{k}} \:\:\:{and}\:\:\lambda_{{k}} =\sum_{{i}+{j}={k}} \:{a}_{{i}} \:{b}_{{j}} \\ $$$$=\:\:\sum_{{i}=\mathrm{0}} ^{{i}={k}} \:\:{a}_{{i}} \:{b}_{{k}−{i}} \:\:=\:\:\sum_{{i}=\mathrm{0}} ^{{i}={k}} \:\:{C}_{{n}} ^{{i}} \:{C}_{{n}} ^{{k}−{i}} \\ $$$$−−>\lambda_{{n}} =\:\sum_{{i}=\mathrm{0}} ^{{i}={n}} \:{C}_{{n}} ^{{i}} \:\:{C}_{{n}} ^{{n}−{i}} \:\:=\sum_{{i}=\mathrm{0}} ^{{i}={n}} \:\left(\:{C}_{{n}} ^{{i}} \right)^{\mathrm{2}} \\ $$$${but}\:\lambda_{{n}} \:{is}\:{the}\:{coefficient}\:{of}\:{x}^{{n}} −−>\:\lambda_{{n}} ={C}_{\mathrm{2}{n}} ^{{n}} \\ $$$${A}_{{n}} =\:{C}_{\mathrm{2}{n}} ^{{n}} \:\:=\:\:\frac{\left(\mathrm{2}{n}\right)!}{\left({n}!\right)^{\mathrm{2}} }\:\:. \\ $$
Answered by Tinkutara last updated on 21/Dec/17
Commented by abdo imad last updated on 21/Dec/17
$${a}\:{lots}\:{of}\:{calculus}\:{but}\:{the}\:{method}\:{is}\:{true}\:.{you}\:{can} \\ $$$${use}\:\Sigma\:{its}\:{more}\:{easy}...{thanks}.... \\ $$