# An Arithmetic Sequence in an Ugly Series

Algebra Level 5

Let $$a_{n} = 4 - 3n,$$ for all integers $$n \geq 1.$$ Define

$f(x, y) = x + \sum_{i=1}^{\infty}\left [\left(\frac{\prod_{j=1}^{i}a_{j}}{3^i\cdot i!} \right )x^{a_{i+1}}y^{i}\right]$

for all real numbers $$x$$ and $$y.$$ Determine the positive integer $$k$$ such that $$f(19, k) = 20.$$

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