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 $(0,0,1,2,3,4,5,6,.......)$