As part of working on a problem, I have encountered the following:
From $\frac{n!}{2!}$, I have to pick $k$ combinations of subfactorials and then sum their squares. The number of combinations will be ${n-2 \choose k}, k=1,2,3...$ I am interested in series for simultaneously increasing $n$ ($>2)$ and $k$, i.e. the next one in the series is ${n+1-2 \choose k+1}$ combinations out of $\frac{(n+1)!}{2!}$.
Example:
$n=5$, so choose from $5\cdot4\cdot3$:
- all combinations of length $1$. This gives a sum of ${3 \choose 1}=3$ squared combinations:
$$\scriptsize 5^2+ 4^2+ 3^2=50$$
$n=6$, so choose from $6\cdot5\cdot4\cdot3$:
- all combinations of length $2$. This gives a sum of ${4 \choose 2}=6$ squared combinations:
$$\scriptsize (6\cdot5)^2+ (6\cdot4)^2+ (6\cdot3)^2+ (5\cdot4)^2+ (5\cdot3)^2+(4\cdot3)^2=2569$$
$n=7$, so choose from $7\cdot6\cdot5\cdot4 \cdot 3$:
- all combinations of length $3$. This gives a sum of ${5 \choose 3}=10$ squared combinations:
$$\scriptsize (7\cdot6\cdot5)^2+ (7\cdot6\cdot4)^2+ (7\cdot6\cdot3)^2+ (7\cdot5\cdot4)^2+ (7\cdot5\cdot3)^2+ (7\cdot4\cdot3)^2+ (6\cdot5\cdot4)^2+ (6\cdot5\cdot3)^2+ (6\cdot4\cdot3)^2+ (5\cdot4\cdot3)^2=157165$$
Etc.
Things are getting complex rather quickly and Sloane's integer sequences didn't reveal any clue.
Could a closed form for such a problem exist?
Thanks!