Coefficient of expansion and limits

When x=1 , coefficient of x^2 = 2c2 = 1
When x=2 , coefficient of x^1 = 3c2 = 3
When x=3 , coefficient of x^0 = 4c2 = 6
When x=4 , coefficient of x^-1 = 5c3 = 10
When x=5 , coefficient of x^-2 = 6c4 = 15
When x=6 , coefficient of x^-3 = 7c5 = 21

Here you can see that for $x = num$ , the coefficient is sum of first $num$ natural numbers.
So when x = 7 , the coefficient $= 8c6 = 8*7/2 = 28$

General term for the reciprocal $\frac{1} A_{n} = \frac{1} {n(n+1)/2}$
General term $\frac{1} A_{n} = 2( \frac{1}{n} - \frac{1}{n+1} )$

The sum = $2( ( \frac{1}{1} - \frac{1}{2} ) + ( \frac{1}{2} - \frac{1}{3} ) + ( \frac{1}{3} - \frac{1}{4} ) + \ldots - \frac{1}{ \infty } )$ = $2*1$