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Working with formal power series

This is a simplified version of my previous question.

1) Is it possible to define a formal power series in sage by giving an expression for the n-th coefficient, e.g. as the expression "n" defines the power series 0 + 1 x + 2 x^2 + 3 x^3 + ... n x^n + ... ?

2) Does sage know how to multiply such objects by convolving the terms? Can it anti-differentiate them symbolically?