${\left( {1 + \frac{1}{x}} \right)^n} = {C_0} + {C_1}.\frac{1}{x} + {C_2}.\frac{1}{{{x^2}}} + ... + {C_n}\left( {\frac{1}{{{x^n}}}} \right)$
on multiplying both expansions, we get
$\frac{{{{(1 + x)}^{2n}}}}{{{x^n}}} = \sum {C_0^2 + x\sum {{C_0}{C_1} + {x^2}\sum {{C_0}{C_2} + ....} } } $$ + {x^r}\sum {{C_0}{C_r} + .....} $
The various sigma are the coefficient of ${x^0},x,{x^2},.....,{x^r}$ in $L.H.S.$ $\frac{{{{(1 + x)}^{2n}}}}{{{x^n}}}$ or coefficient of ${x^n},{x^{n + 1}},{x^{n + 2}},.....,{x^{n + r}}$ in the expansion of ${(1 + x)^{2n}}$ which occur in ${T_{n + 1,}}{T_{n + 2}},....$ and are
$^{2n}{C_n}{,^{2n}}{C_{n + 1}}{,^{2n}}{C_{n + 2}}{....^{2n}}{C_{n + r}}$etc.
$^{\,2n}{C_{n + 2}} = \frac{{(2\,n\,)\,!}}{{(n - 2)\,!\,(n + 2)\,!}}$
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