These sums are getting out of hand

Discrete Mathematics Level 5

Let \(\mathbb{Z}^*\) be the set of non-negative integers.

Given that \(\lambda_0, \lambda_1, \lambda_2, \ldots\) are real numbers that satisfy, for all \(n \in \mathbb{Z}^*\),

\[\displaystyle \sum_{a+b+c=n} \lambda_a \lambda_b \lambda_c = 1\]

where the sum is taken over all \(a,b,c \in\mathbb{Z}^*\) such that \(a+b+c = n\). (In particular, \(\lambda_0 = 1\).)

It turns out that for all \(n \in \mathbb{Z}^*\) there is a nice closed form for \(\lambda_n\). Using such a form, determine \(\lfloor 1000 \lambda_{2016}\rfloor\).


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