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Factorisation by Regrouping Terms

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Question

Find the generating functions of the series $1 + 2 x + 9 x^{2} + 20 x^{3} + 35 x^{4} + 54 x^{5} + . . . . .$

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Solution

Let $S = 1 + 2 x + 9 x^{2} + 20 x^{3} + 35 x^{4} + 54 x^{5} + \dots$
$\Rightarrow - 3 x S = - 3 x - 6 x^{2} - 27 x^{3} - 60 x^{4} - 105 x^{5} - \dots$
$\Rightarrow 3 x^{2} S = 3 x^{2} + 6 x^{3} + 27 x^{4} + 60 x^{5} + \dots$
$\Rightarrow - x^{3} S = - x^{3} - 2 x^{4} - 9 x^{5} - \dots$
Adding the above series, we get
$(1 - 3 x + 3 x^{2} - x^{3}) S = 1 - x + 6 x^{2} - 2 x^{3}$
$\Rightarrow (1 - x)^{3} S = 1 - x + 6 x^{2} - 2 x^{3}$
$\Rightarrow S = \frac{1 - x + 6 x^{2} - 2 x^{3}}{(1 + x)^{3}}$

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