Calculus Level 5

$2\sum_{k=1}^{\infty}k^2f^{(k)}(x)=3f(x),\hspace{10mm}f(0)=2016$

Find the function $$f:\mathbb{R}\rightarrow\mathbb{R}$$ that satisfies the differential equation above for all real $$x$$. Return $$f'(0)$$ as your answer.

If you come to the conclusion that no such function exists (or there is more than one), enter 666.

Clarification: $$f^{(k)}(x)$$ denotes the $$k^\text{th}$$ derivative of $$f(x)$$.

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