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Please tell me the approach of solving the question for finding a closed formula for the given generating function:-

$x^{2} / (1-x)^{2}$ . Please determine the general approach how to solve
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Generating function:-Suppose that generating sequence given that $(a_{0},a_{1},a_{2},a_{3},\cdot\cdot\cdot\cdot\cdot\cdot\cdot)$

$G(x) =a_{0}x^{0}+a_{1}x^{1}+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+...............$ 

$G(x) =a_{0}+a_{1}x^{1}+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+...............   --->(1)$

Let generating sequence are $(1,1,1,1,1,1,1,1,1,1,1,............)$

So, Generating function is:

$G(x)=1.x^{0}+1.x^{1}+1.x^{2}+1.x^{3}+1.x^{4}+1.x^{5}+1.x^{6}+..............$

$G(x)=1+1.x^{1}+1.x^{2}+1.x^{3}+1.x^{4}+1.x^{5}+1.x^{6}+..............$

$G(x)=1+x^{1}+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}+..............  ------>(2)$

From the equation$(2)$,we apply some operation on generating function

$1+x^{1}+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}+..............   \Leftrightarrow  \frac{1}{1-x}$   [Sum of infinite series]  

Differentiate both side with respect to $'x'$

$\frac{\mathrm{d} }{\mathrm{d} x}[1+x^{1}+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}+..............]   \Leftrightarrow  \frac{\mathrm{d} }{\mathrm{d} x}[\frac{1}{1-x}]$ 

 $\frac{\mathrm{d} }{\mathrm{d} x}[1+x^{1}+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}+..............]\Leftrightarrow \frac{\mathrm{d} }{\mathrm{d} x}[(1-x)^{-1}]$

$[0+1x^{0}+2x^{1}+3x^{2}+4x^{3}+5x^{4}+6x^{5}+..............]\Leftrightarrow[(-1)(1-x)^{-2}(-1)]$

$[1x^{0}+2x^{1}+3x^{2}+4x^{3}+5x^{4}+6x^{5}+..............]\Leftrightarrow[(1-x)^{-2}]$

$[1+2x+3x^{2}+4x^{3}+5x^{4}+6x^{5}+..............]\Leftrightarrow[\frac{1}{(1-x)^{2}}]$

Multiply both side with $x^{2}$

$[x^{2}+2x^{3}+3x^{4}+4x^{5}+5x^{6}+6x^{7}+..............]\Leftrightarrow[\frac{x^{2}}{(1-x)^{2}}]$

$[0x^{0}+0x^{1}+1x^{2}+2x^{3}+3x^{4}+4x^{5}+5x^{6}+6x^{7}+..............]\Leftrightarrow[\frac{x^{2}}{(1-x)^{2}}]$

So This is the closest form of the given generating function.

and generating sequence are $(0,0,1,2,3,4,5,6,.......)$

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