Is it possible to simplify the sum $\sum\limits_{n = 0}^{\lfloor {\frac{M}{2}}\rfloor } {\frac{{M!}}{{{2^n} \times ( {n!} )\times( {M - 2n})!}}}$?

Question

Is it possible to further simplify the expression

$$\sum\limits_{n = 0}^{\left\lfloor {\frac{M}{2}} \right\rfloor } {\frac{{M!}}{{{2^n} \times \left( {n!} \right) \times \left( {M - 2n} \right)!}}},$$

where $M$ is a positive integer?

Answer 1

Making the problem more general $$S_M=\sum\limits_{n = 0}^{\left\lfloor {\frac{M}{2}} \right\rfloor } {\frac{{M!}}{{ {n!} \, \left( {M - 2n} \right)!}}}x^n$$ $$S_{2m}=(-1)^m (4x)^m \,U\left(-m,\frac{1}{2},-\frac{1}{4 x}\right)$$ $$S_{2m+1}=(-1)^m (4x)^m \,U\left(-m,\frac{3}{2},-\frac{1}{4 x}\right)$$ where appears the confluent hypergeometric function.

If $x=\frac 12$, these generate the sequences $$\{2,10,76,764,9496,140152,2390480,46206736,997313824,23758664096\}$$ which correspond to $$S_{2m}=2^n n!\, L_n^{-\frac{1}{2}}\left(-\frac{1}{2}\right)$$ and $$\{4,26,232,2620,35696,568504,10349536,211799312,4809701440,119952692896\}$$ which correspond to $$S_{2m+1}=2^n n!\, L_n^{\frac{1}{2}}\left(-\frac{1}{2}\right)$$ where appear the generalized Laguerre polynomials.

Have a look at $OEIS$ for very interesting informations.