MHB Find a closed-form expression for f(n)

[alexmahone]

Find a closed-form expression for $f(n)$ where

$f(n)=1+n+n(n-1)+n(n-1)(n-2)+...+n(n-1)\cdots2$ for $n>1$

and $f(1)=1$.

(So, $f(2)=1+2=3$

$f(3)=1+3+3\cdot 2=10$

$f(4)=1+4+4\cdot 3+4\cdot 3\cdot 2=41$

and so on.)

 

[Evgeny.Makarov]

It's easy to convert this into a recurrence relation, but I doubt that a simple closed form exists.

Spoiler

See OEIS and WolframAlpha.

Spoiler

$$f(n)=n!\left(1+\frac{1}{2!}+\dots+\frac{1}{n!}\right)$$

 

[alexmahone]

It's easy to convert this into a recurrence relation, but I doubt that a simple closed form exists.

I actually started with the recurrence relation $f(n)=1+nf(n-1)$ to arrive at my formula in post #1.