Resistor for 20 LEDs in series
When you connect the LEDs in series, the forward voltages of the LEDs add up. If you have to connect 20 LEDs in series, the forward voltage of the string of LEDs is 20 * 3.8V = 76V. But then you only need 20mA per series of LEDs! 
(You got that backwards)



If you instead contact 20 LEDs in parallel, the forward voltage is 3.8V, but now the current is 20 * 20mA. You now can use a 5V supply (since 5V > 3.8V). 
In that case, you need a resistor in front of every LED to "take care" of the difference of your 5V supply and the 3.8V LED's forward voltage (5V - 3.8V = 1.2V), that's what all the LED-Resistor-Calculators show you.

So, to make LEDs light up in series, you need to supply 20mA, independent of the number of LEDs. An LED driver does that. It supplies 20mA and "adjusts" the voltage to make that happen. You normally would not put 20 LEDs in series, since your LED driver had to supply ~76V (which the power supply has to supply in the first place).

A 5V supply can do 1 LED in series (5V > 3.8V).
A 12V supply can do 3 LEDs in series (12V > 3*3.8V).
For every string of LEDs you than provide 20mA each.