# Ackerman function 2

Algebra Level pending

Let's reiterate about Ackerman function $$\mathcal{A}$$.

$$\mathcal{A}(m,n) = \begin{cases} n+1 & \text{if } m = 0 \\ \mathcal{A}(m-1, 1) & \text{if } m > 0, n = 0 \\ \mathcal{A}(m-1, \mathcal{A}(m-1, n-1)) & \text{if } m > 0, n > 0 \end{cases}$$

Unlike what's suspected, this function actually doesn't grow as quickly as Ackermann function!

Let $$\displaystyle S(n) = \sum_{i=0}^n \sum_{j=0}^n \mathcal{A}(i,j)$$. It turns out that $$S(n)$$ is a polynomial for all sufficiently large $$n$$; that is, there exists some integer $$N$$ and a polynomial $$P$$ such that for all integer $$n > N$$, $$S(n) = P(n)$$. The sum of the coefficients of $$P(n)$$ can be expressed as $$\frac{a}{b}$$ for some positive integers $$a,b$$ that are relatively prime. Compute $$a+b$$.

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