I plugged a frequency response of $\omega$ into the inverse discrete-time Fourier transform (inverse DTFT) equation and ended up with:

$$h[k] = \frac{sin(k\omega_c)}{k^2\pi}-\frac{(\omega_c)cos(k\omega_c)}{k\pi}$$

where $\omega_c$ is the cutoff freq measured in radians. Index $k$ is ...-3,-2,-1,0,1,2,3.... When $k = 0$ set $h[0] = 0$. (You may want to multiply $h[k]$ by a window sequence to reduce the frequency magnitude response ripples.) If we set $\omega_c = \pi$ we end up with:

$$h[k] = \frac{-(-1)^k}{k}$$

Try out the above $h[k]$'s. Who knows, they might work for you. (The $h[k]$ equations are supplied "as is." No warranty of any kind, either expressed or implied.)

I plugged a frequency response of $\omega$ into the inverse discrete-time Fourier transform (inverse DTFT) equation and ended up with:

$$h[k] = \frac{\sin(k\omega_c)}{k^2\pi}-\frac{(\omega_c)\cos(k\omega_c)}{k\pi}$$

where $\omega_c$ is the cutoff freq measured in radians. Index $k$ is ...-3,-2,-1,0,1,2,3.... When $k = 0$ set $h[0] = 0$. (You may want to multiply $h[k]$ by a window sequence to reduce the frequency magnitude response ripples.) If we set $\omega_c = \pi$ we end up with:

$$h[k] = \frac{-(-1)^k}{k}$$

Try out the above $h[k]$'s. Who knows, they might work for you. (The $h[k]$ equations are supplied "as is." No warranty of any kind, either expressed or implied.)