27020

27020 (Gheorghe Szöllösy)

Să se calculeze suma $\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\frac {1}{4^{k}\cdot (k!)^{2}\cdot (n-2k)!}},\quad n\geq 1$

Soluție:

Fie $a_{n}$ coeficientul lui $X^{n}$ din rezolvarea lui

$$P(X)=\left(X+\left\lfloor {\frac {1}{2}}\right\rfloor \right)^{2n}=\left(X(1+X)+\left\lfloor {\frac {1}{4}}\right\rfloor \right)^{n}=\sum _{k=0}^{n}C_{n}^{k}X^{(n-k)}\left({\frac {1}{4^{k}}}\right).$$

Avem $a_{n}=\left({\frac {1}{2^{n}}}\right)C_{2}n^{n}$, iar pe de altă parte, Failed to parse (syntax error): {\displaystyle a_n = C_n^0 \cdot C_n^0 + C_n^1 \cdot C_(n-1)^1 \left.\frac{1}{4}\right + C_n^2 \cdot C_(n-2)^1\left(\frac{1}{4^2}\right) + ... = } Failed to parse (syntax error): {\displaystyle = \sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor} C_n^k C_(n-k)^k \cdot \left.\frac{1}{4^k}\right = n! \sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor} \frac{1}{(k!)^2 (n-k)! 4^k},}