Factorial Sum

Sums with unity Numerator and Factorials in the Denominator which can be expressed analytically include

$$\sum_{i=1}^n {1\over (n+i-k)!(n-i)!} = {{}_2F_1(1,-n;1+n-k;-1)-1\over\Gamma(1+n)\Gamma(1+n-k)}$$ (1)

$$\sum_{i=1}^n {1\over (n+i-1)!(n-i)!} = {n\sqrt{\pi}\over 2\Gamma({\textstyle{1\over 2}}+n)\Gamma(1+n)}$$ (2)

$$\sum_{i=1}^n {1\over (n+i)!(n-i)!} = {\sqrt{\pi}\over 2\Gamma({\textstyle{1\over 2}}+n)\Gamma(1+n)}-{1\over 2\Gamma^2(1+n)}$$ (3)

$$\sum_{i=1}^n {1\over (n+i+1)!(n-i)!} = {\sqrt{\pi}\over 2\Gamma({\textstyle{3\over 2}}+n)\Gamma(1+n)}-{1\over 2\Gamma(1+n)\Gamma(2+n)},$$ (4)

where ${}_2F_1(a,b;c;z)$ is a Hypergeometric Function and $\Gamma(z)$ is a Gamma Function.

Sums with $i$ in the Numerator having analytic solutions include

$$\sum_{i=1}^n {i\over (n+i-k)!(n-i)!} = {n\,{}_2F_1(2,1-n;2-k+n;-1)\over (1-k+n)\Gamma(1+n)\Gamma(1-k+n)}$$ (5)

$$\sum_{i=1}^n {i\over (n+i-1)!(n-i)!} = {1\over 2\Gamma(n)} \left[{{\sqrt{\pi}\over 2\Gamma({\textstyle{1\over 2}}+n)}+{n\over\Gamma(1+n)}}\right]$$ (6)

$$\sum_{i=1}^n {i\over (n+i)!(n-i)!} = {n\over 2\Gamma^2(1+n)}$$ (7)

$$\sum_{i=1}^n {i\over (n+i+1)!(n-i)!} = {1\over 2\Gamma(1+n)}\left[{{1\over\Gamma(2+n)}-{(n^2+3n+2)\sqrt{\pi}\over 2\Gamma({\textstyle{3\over 2}}+n)}\,}\right].$$ (8)

A sum with $i^2$ in the Numerator is

$\sum_{i=1}^n {i^2\over (n+i-k)!(n-i)!}$
$= {n\over (1-k+n)(2-k+n)\Gamma(1+n)\Gamma(1-k+n)} [(2-k+n)\,{}_2F_1(2,1-n;2-k+n;-1)$
$+2(n-1)\,{}_2F_1(3,2-n;3-k+n;-1)],\quad$	(9)

where ${}_2F_1(a,b;c;z)$ is the Hypergeometric Function.

Sums of factorial Powers include

$$\sum_{n=0}^\infty {(n!)^2\over(2n)!}={4\over 3}+{2\pi\over 9\sqrt{3}}$$ (10)

$$\sum_{n=0}^\infty {(n!)^3\over(3n)!}=\int_0^1 [P(t)+Q(t)\cos^{-1} R(t)]\,dt,$$ (11)

where

$$P(t) = {2(8+7t^2-7t^3)\over(4-t^2+t^3)^2}$$ (12)

$$Q(t) = {4t(1-t)(5+t^2-t^3)\over (4-t^2+t^3)^2\sqrt{(1-t)(4-t^2+t^3)}}$$ (13)

$$R(t) = 1-{\textstyle{1\over 2}}(t^2-t^3)$$ (14)

(Beeler et al. 1972, Item 116).

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.