Is there a fast way to compute coefficient of some term of the product of some series'?

Question

The example in wikipedia is

$$A=1-3x+5x^2-7x^3+9x^4-11x^5+\cdots$$

$$B=2x+4x^3+6x^5+\cdots$$

$$AB=2x-6x^2+14x^3-26x^4+44x^5+\cdots$$

And the term $x^5$ is given by

$$44x^5=(1\cdot6x^5)+(5x^2\cdot4x^3)+(9x^4\cdot2x)$$

So if you wanted the 5th term of $AB$, you would get $44$ in this example.

Answer 1

Given $A=\sum_{k=0}^\infty a_kx^k, B=\sum_{k=0} b_kx^k$, their product $AB$ is known as the series $$ C=\sum_{k=0}^\infty c_kx^k, $$ where $c_k=\sum_{j=0}^k a_jb_{k-j}$. Look up Cauchy product for more info.