Impulse response of a system from the input-output relation I need to find out the recurrence relation and the impulse response of the following system.

The relation seems to be $$y[n] = \frac{2}{16} x[n] + \frac{3}{16} x[n-1] + \frac{6}{16} x[n-2] + \frac{3}{16} x[n-3] + \frac{2}{16}x[n-4]$$

Since there is no recursion the impulse response $h[n]$ would contain only 5 values (below) being the coefficients?

[2/16, 3/16, 6/16, 3/16, 2/16]
It looks like you answered your own question but to be complete..The impulse response $H(z)=\frac{Y(z)}{X(z)}$ can be found from the difference equation which you have found from the block diagram. So taking the z-transform of both sides of the difference equation we have: $Y(z)=X(z)(\frac{2}{16}+\frac{3}{16}z^{-1}+\frac{6}{16}z^{-2}+\frac{3}{16}z^{-3}+\frac{2}{16}z^{-4})$ and so we have $H(z)=\frac{2}{16}+\frac{3}{16}z^{-1}+\frac{6}{16}z^{-2}+\frac{3}{16}z^{-3}+\frac{2}{16}z^{-4}$.

Now taking the inverse z-transform you get the impulse response to be $h[n]=\frac{2}{16}+\frac{3}{16}\delta[n-1]+\frac{6}{16}\delta[n-2]+\frac{3}{16}\delta[n-3]+\frac{2}{16}\delta[n-4]$, which has the coefficients that you listed.