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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