The LM3914 can drive up to three red LEDs in series on each output when it has a 9V supply. It can be set to provide a regulated current up to 30mA on each output which is plenty for ordinary LEDs. Its regulated output current does not need current-limiting resistors.
Your circuit with the capacitors would just blink the LEDs for a moment.
The LM3914 can have the 1st LED in series with the 10th LED and they would light and stay lighted, then the 2nd and 9th LEDs would light and stay lighted, then ther 3rd and 8th LEDs would light and stay lighted, then the 4th and 7th ....., exactly what you want.
Or the LM3914 can light up to 10 LEDs or up to 30 LEDs (in series) in sequence.