Generalized Cantor set segment lengths
Jelly, &nbsp;15 13&nbsp; 12 bytes

-2 thanks to Dennis (using a Link rather than a chain allows right to be used implicitly by ¡; No need to wrap the 1 in a list due to the fact that Jelly prints lists of one item the same as the item)
-1 thanks to Erik the Outgolfer (use Ɗ to save the newline from using Ç)

1×€³§JḤ$¦ẎƊ¡


A full program printing a list in Jelly format (so [1] is printed as 1)

Try it online!

How?

1×€³§JḤ$¦ẎƊ¡ - Main link: segmentLengths; iterations
1            - literal 1 (start with a single segment of length 1)
           ¡ - repeat...
             - ...times: implicitly use chain's right argument, iterations
          Ɗ  - ...do: last 3 links as a monad (with 1 then the previous output):
   ³         - (1) program's 3rd argument = segmentLengths
 ×€          -  1  multiply €ach (e.g. [1,2,3] ×€ [1,2,1] = [[1,4,3],[2,4,2],[3,6,3]])
        ¦    -  2  sparse application... 
       $     - (2) ...to: indices: last two links as a monad:
     J       - (2)          range of length = [1,2,3,...,numberOfLists]
      Ḥ      - (2)          double            [2,4,6,...] (note: out-of bounds are ignored by ¦)
    §        - (2) ...of: sum each (i.e. total the now split empty spaces)
         Ẏ   -  3  tighten (e.g. [[1,2,3],4,[5,6,7]] -&gt; [1,2,3,4,5,6,7])
             - implicit print