My Eleventh Problem

Algebra Level 4

Let \(n\in\mathbb{N}\) and \(f(0)=1\).

Simplify \(f\) for:

\(f(n)=k^{n}+\sum_{a=1}^{n}[mk^{a-1}\times(k^{n-a}+\sum_{b=1}^{n-a}[mk^{b-1}\times(k^{n-a-b}+\sum_{c=1}^{n-a-b}[mk^{c-1}\times(k^{n-a-b-c}+\sum_{d=1}^{n-a-b-c}[... ...])])])]\)

for arbitrary constants \(k\) and \(m\), where \(f\) is a function of a natural number or \(0\).

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