 Closed figure succession for reciprocal cubic duty Answer

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Closed figure succession for reciprocal cubic duty

deem a dice of the figure f (x) =$$x ^ 3-2x + z$$

Is it workable to emanate an influence succession of coefficients for the duty $$x ^ y / f (x)$$, for some $$y = 0,1,2 …$$ this doesn’t require the employ of Faà di Bruno’s components or it doesn’t require a number of iterations of a recursive strategy to get the x ^ n coefficients.

for instance:

The sixth coefficient is $$((32 – 12 z ^ 2)) / z ^ 6$$

So how do you emanate such features from z with out having to employ a cumbersome recursive strategy?

It is workable to emanate such a closed expression for the coefficients of $$frac {3x ^ 2-2} {x ^ 3-2x + z}$$

$$displaystyle -z ^ {- m} 2 ^ {m + 1} + sum_ {n = 0} frac {z ^ {2-m} z ^ {2n} (1 + m)} {2 ^ { 3n-m + 2} (n + 1)} binom {3n + 1-m} {n}$$

to the $$3n + 1-m <0$$