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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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) } .