I want to extract some informations, say the coefficients of $x^{500}$ and $x^{1000}$ terms (numerical values are acceptable), of the multiple of
(2/9 x^(-1) + 5/9 + 2/9 x)^700*(1/3 x^(-1) + 1/3 + 1/3 x)^600
However, the brutal computation would not be a practical way. I think the underlying idea is related to the convolution, but I can't see a way to do this. It would be good to have a lazy evaluation mechanism in MMA, but there seems no.
Actually this question comes from a probability task, that I want to compute the p.d.f. of the sum of, say 700 many random variables.
k = 700;
vec = Array[x, k];
dist = EmpiricalDistribution[{2/9, 5/9, 2/9} -> {-1, 0, 1}];
pdf = PDF@
TransformedDistribution[Total[vec],
vec \[Distributed] ProductDistribution[{dist, k}]]
DiscretePlot[pdf[n], {n, -10, 10}, PlotRange -> All]
(*It took endless time... Hence not work.*)
How can I achieve this?
PS. If it is inevitable to neglects some intermediate terms with the norm of coefficients less than, say $2^{-300}$, to achieve the task, then I'm willing to do so. But how to do it?
Coefficient
returns answers quite quickly $\endgroup$