Kernel panic and unable to boot Ubuntu 16.04 after updating

You are missing the initramfs for kernel version 4.8.0-44.

It looks like you got a kernel panic error message after updating the kernel in Ubuntu 16.04.1 to 4.8.0-44-generic. To fix it enter a text-only console and either replace 4.8.0-44-generic with the more stable default kernel version or run sudo update-initramfs -u -k 4.8.0-44-generic && sudo update-grub.

1. After the boot stops at the black screen, open a text-only virtual console by using the keyboard shortcut Ctrl+Alt+F3

2. At the login: prompt type your username and press Enter.

3. At the Password: prompt type your user password and press Enter.

4. After you have logged in in the console type:

   sudo apt remove linux-image-4.8.0-44-generic linux-image-extra-4.8.0-44-generic linux-headers-4.8.0-44-generic    
   sudo apt install linux-image-4.4.0-71-generic linux-headers-4.4.0-71 linux-headers-4.4.0-71-generic 
   sudo reboot
   

   Alternatively you can run the following commands to update initramfs for kernel version 4.8.0-44-generic :

   sudo update-initramfs -u -k 4.8.0-44-generic
   sudo update-grub
   

5. Immediately after the BIOS/UEFI splash screen during boot, with BIOS, quickly press and hold the Shift key, which will bring up the GNU GRUB menu. (If you see the Ubuntu logo, you've missed the point where you can enter the GRUB menu.) With UEFI press (perhaps several times) the Esc key to get to the GRUB menu. Sometimes the manufacturer's splash screen is a part of the Windows bootloader, so when you power up the machine it goes straight to the GRUB screen, and then pressing Shift is unnecessary.

6. From the purple GRUB screen select Advanced options for Ubuntu with the ↑ and ↓ keys and press Enter.

7. A new purple screen will appear showing a list of kernels. Select Ubuntu, with Linux 4.4.0-71-generic and press Enter.

8. Ubuntu will load the selected kernel and proceed to the login screen as usual.

9. Don't let Ubuntu update the kernel to the 4.8.0-44 version that caused the black screen again, or else you'll get the same black screen problem that you got before.

Steps:

1. Turn off the system completely. 
2. Turn system back on.
3. Immediately after the System Manufacture Logo or Boot Message Press Shift to goto Grub options. Or You can force kill the system with power button and turn it back on, the Grub menu should appear.
4. Choose Advance option For Ubuntu.
5. A selection menu should appear with list of kernels installed on the system, take note of the version number of the latest kernel ( Usually the First one) choose the recovery option of the  old kernel (one with lower version number of the first one, normally the fourth 4th option).
6. Another list of options should appear, choose root : Drop to to root shell option, if you set root account password enter that here else press enter.

7. Execute the following commands:  

   mount -o remount,rw /
   mkinitramfs -o /boot/initrd.img-{kerner_version}-generic {kernel_version}-generic
   update-grub
   

   Here {Kernel_version} is the kernel version number you take note of in step 5 (The Problematic kernel, latest installed one).

Reboot the system that is it, you're all set and ready to continue your work. Hope that helps.