Series expansion with criteria on the coefficients

Question

I will first do an illustrative example. Suppose I have the following function:

$ f(\vec{x},\vec{t})=\frac{x_1x_2}{(1-x_1 x_2^{-1} t_1)(1-x_2x_1^{-1} t_2)}$

I want to expand it with respect to $(t_1,t_2)$, and then select only those terms that are proportional to $x_1,x_2$. So, basically, in my example, I can do

Coefficient[Expand[Normal[Series[x[1] x[2]/((1 - h t[1] x[1] x[2]^-1) (1 - h t[2] x[2] x[1]^-1)), {h, 0, 10}]] /. h -> 1], x[1] x[2]]

And the result will be

1 + t[1] t[2] + t[1]^2 t[2]^2 + t[1]^3 t[2]^3 + t[1]^4 t[2]^4 + t[1]^5 t[2]^5

Now, the problem comes when there are very complicated functions, depending on an arbitrary number of variables $\vec{t}$, so the series is very slow at higher powers because of the large coefficients that may be present. There is also a problem of Expand that will saturate the RAM eventually, and if I don't do the Expand, the Coefficient is not selected properly.

Is there a way to make the Series Expansion and at the same time select the coefficients satisfying a given criterion for the coefficients at the same time?

Here a more complicated example:

$\frac{x_{1} x_{2} x_{3} x_{4} x_{5} x_{6} x_{7} x_{8} x_{9} x_{10} x_{11} x_{12} x_{13}}{\left(1-\frac{t_{1}}{x_{1} x_{2}}\right) \left(1-\frac{t_{2}}{x_{1} x_{2}}\right) \left(1-\frac{t_{9} x_{1} x_{2}}{x_{4}}\right) \left(1-\frac{t_{3}}{x_{3} x_{4}}\right) \left(1-\frac{t_{4}}{x_{3} x_{4}}\right) \left(1-\frac{t_{11} x_{3} x_{4}}{x_{2}}\right) \left(1-\frac{t_{5}}{x_{5}}\right) \left(1-\frac{t_{6}}{x_{5}}\right) \left(1-\frac{t_{12} x_{2} x_{7}}{x_{9}}\right) \left(1-\frac{t_{7} x_{5} x_{9}}{x_{8}}\right) \left(1-\frac{t_{10} x_{4} x_{6}}{x_{10}}\right) \left(1-\frac{t_{8} x_{5} x_{10}}{x_{11}}\right) \left(1-\frac{t_{13} x_{4} x_{8}}{x_{12}}\right) \left(1-\frac{t_{14} x_{3} x_{12}}{x_{6}}\right) \left(1-\frac{t_{15} x_{2} x_{11}}{x_{13}}\right) \left(1-\frac{t_{16} x_{1} x_{13}}{x_{7}}\right)}$

and I would like to select only those terms that are multiplied by $x_{1} x_{2} x_{3} x_{4} x_{5} x_{6} x_{7} x_{8} x_{9} x_{10} x_{11} x_{12} x_{13}$

Answer 1

My idea is as follows.

Start from a given function

f = x[1] x[2]/((1 - h t[1] x[1] x[2]^-1) (1 - h t[2] x[2] x[1]^-1))

and discard the prefactor. Thus we a looking for the terms in the expansion that do not contain x[i]

g = 1/((1 - (h t[1] x[1])/x[2]) (1 - (h t[2] x[2])/x[1])) /. {x[i_] ->y^(1 + 10 i)}
Coefficient[Normal[Series[g, {h, 0, 10}]] /. h -> 1, y, 0]
(* 1 + t[1] t[2] + t[1]^2 t[2]^2 + t[1]^3 t[2]^3 + t[1]^4 t[2]^4 + t[1]^5 t[2]^5*)

The replacement above allows to reduce many variables x[i] to a single one y. Hopefully it will speed up the things.