Question

Find the generating functions of the series $1+3x+7x^2+13x^3+21x^4+31x^5+.....$

Answer 1

Let $S=1+3x+7x^2+13x^3+21x^4+31x^5+\dots$

$\Rightarrow -3xS=-3x-9x^2-21x^3-39x^4-63x^5-\dots$

$\Rightarrow 3x^{2}S=3x^2+9x^3+21x^4+39x^5+\dots$

$\Rightarrow -x^{3}S=-x^3-3x^4-7x^5-\dots$

Adding the above series, we get

$(1-3x+3x^2-x^3)S=1+x^2$

$\Rightarrow (1-x{)}^{3}S=1+x^2$

$\Rightarrow S=\frac{1-x^2}{(1+x{)}^3}$