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Division of a Polynomial by a Monomial

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Question

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

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Solution

$S = 1 + 16 x + 81 x^{2} + 256 x^{3} + 625 x^{4} + 1296 x^{5} \dots$

$- 5 x S = - 5 x - 80 x^{2} - 405 x^{3} - 1280 x^{4} - 3125 x^{5} - \dots$

$10 x^{2} S = 10 x^{2} + 160 x^{3} + 810 x^{4} + 2560 x^{5} + \dots$

$- 10 x^{3} S = - 10 x^{3} - 160 x^{4} - 810 x^{5} - \dots$

$5 x^{4} S = 5 x^{4} + 80 x^{5} + \dots$

$- x^{5} S = - x^{5} - \dots$

Adding the above series, we get

$(1 - 3 x + 10 x^{2} - 10 x^{3} + 5 x^{4} - x^{5}) S = 1 + 11 x + 11 x^{2} + x^{3} ⟹ (1 - x)^{5} S = 1 + 11 x + 11 x^{2} + x^{3} ⟹ S = \frac{1 + 11 x + 11 x^{2} + x^{3}}{(1 - x)^{5}}$

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