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working with coefficients of formal series

I want to define a truncated series or polynomial of arbitrary degree and then work algebraically with the polynomial to solve for various quantities in terms of the coefficients. When I write something like

i,n,z=var('i,n,z')
c=function('c')
p=z^(-4)+sum(c(i)/z^i,i,0,2)
p

this returns

(z^2*c(0) + z*c(1) + c(2))/z^2 + 1/z^4

But if I try to define an arbitrary polynomial of this type

p(n)=z^(-2n) + sum(c(i)/z^i,i,0,2n-2)
p(2)

returns

z^4 + sum(z^(-i)*c(i), i, 0, 2)

What is the crucial difference here?

I had a similar problem when working with truncated power series over the ring PowerSeriesRing(SR). I want to manipulate these expressions algebraically as elements of a ring then ask for info about certain coefficients. But working with formal sums and substituting values of n returns power series coefficients, e.g. the following expression as the coefficient of z^-4 (after some computations...f and g are symbolic functions)

1/4*(sum(z^i*f(i), i, 1, 3)*sum(z^i*g(i), i, 1, 3) + 2)^2 + sum(z^i*f(i), i, 1, 3)*sum(z^i*g(i), i, 1, 3)

How do I get sage to work with the series and also give info about the coefficients, i.e. multiply series expressions but then expand them out in z? I've tried expand() in this setting with mixed success.