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Factorisation by Regrouping Terms

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Question

Find the generating functions of the series $2 + 5 x + 10 x^{2} + 17 x^{3} + 26 x^{4} + 37 x^{5} + . . . .$

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Solution

$S = 2 + 5 x + 10 x^{2} + 17 x^{3} + 26 x^{4} + 37 x^{5} + \dots$

$- 3 x S = - 6 x - 15 x^{2} - 30 x^{3} - 51 x^{4} - 78 x^{5} - \dots$

$3 x^{2} S = 6 x^{2} + 15 x^{3} + 30 x^{4} + 51 x^{5} + \dots$

$- x^{3} S = - 2 x^{3} - 5 x^{4} - 10 x^{5} - \dots$

Adding the above series, we get

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

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