Repeated indefinite integral

Calculus Level 5

Let

\(\begin{align*}f(n)&=\displaystyle\underbrace{\iint\cdots\int}_{n\text{ times}} \underbrace{dxdx\cdots dx}_{n\text{ times}}\\ &=\dfrac{x^n}{a_0}+\dfrac{c_1x^{n-1}}{a_1}+\dfrac{c_2x^{n-2}}{a_2}+\cdots +\dfrac{c_n}{a_n}\end{align*}\)

where \(a_i\) and \(c_i\) are constants left over from performing the integration. Let \(g(n)=\displaystyle\sum^n_{i=0}{a_i}\) where \(a_i\) are the constants in \(f(n).\) Find the last three digits of \(g(10).\)

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