I have to solve a the following recurrence through iteration. I also show the steps. The issues is I have no idea how to get to the closed form! I can tell there's a pattern, but I don't know how to "condense" it into the closed form.
\begin{align*} T(n) = nT(n - 1) + 1\\ T(n) &= n((n - 1)T(n - 2) + 1) + 1\\ T(n) &= n(n - 1)T(n - 2) + n + 1 \\ T(n) &= n(n-1)(n-2)T(n-3) + n(n-1) + n + 1 \\ T(n) &= n(n-1)(n-2)(n-3)T(n-4) + n(n-1)(n-2) + n(n-1) + n + 1 \end{align*}
Some direction or explanation would be greatly appreciated!!! Thanks :)
In your previous post, your received nice answers.
Continuing in the same spirit and trying to find a closed form, the given results can "simplify" to $$T(n)=(c-1)\, n!+e \,\Gamma (n+1,1)$$ where appears the incomplete gamma function $(c=T(0))$.