Seven numbers, each 1 or -1, are listed in a row in such a way that adding the numbers successively from left to right never gives a negative answer. How many valid lists are there?

We can enumerate the valid sequences as follows:

Noting that there may be at most three #-1's#, we can write down the possibilities hierarchically. For each valid sequence with less than three #-1#'s, write out nested possibities below it with #-1#'s in each possible position that occurs after the existing #-1#'s:

#+1, +1, +1, +1, +1, +1, +1#

#+1, color(red)(-1), +1, +1, +1, +1, +1#

#+1, color(red)(-1), +1, color(red)(-1), +1, +1, +1#

#+1, color(red)(-1), +1, color(red)(-1), +1, color(red)(-1), +1#

#+1, color(red)(-1), +1, color(red)(-1), +1, +1, color(red)(-1)#

#+1, color(red)(-1), +1, +1, color(red)(-1), +1, +1#

#+1, color(red)(-1), +1, +1, color(red)(-1), color(red)(-1), +1#

#+1, color(red)(-1), +1, +1, color(red)(-1), +1, color(red)(-1)#

#+1, color(red)(-1), +1, +1, +1, color(red)(-1), +1#

#+1, color(red)(-1), +1, +1, +1, color(red)(-1), color(red)(-1)#

#+1, color(red)(-1), +1, +1, +1, +1, color(red)(-1)#

#+1, +1, color(red)(-1), +1, +1, +1, +1#

#+1, +1, color(red)(-1), color(red)(-1), +1, +1, +1#

#+1, +1, color(red)(-1), color(red)(-1), +1, color(red)(-1), +1#

#+1, +1, color(red)(-1), color(red)(-1), +1, +1, color(red)(-1)#

#+1, +1, color(red)(-1), +1, color(red)(-1), +1, +1#

#+1, +1, color(red)(-1), +1, color(red)(-1), color(red)(-1), +1#

#+1, +1, color(red)(-1), +1, color(red)(-1), +1, color(red)(-1)#

#+1, +1, color(red)(-1), +1, +1, color(red)(-1), +1#

#+1, +1, color(red)(-1), +1, +1, color(red)(-1), color(red)(-1)#

#+1, +1, color(red)(-1), +1, +1, +1, color(red)(-1)#

#+1, +1, +1, color(red)(-1), +1, +1, +1#

#+1, +1, +1, color(red)(-1), color(red)(-1), +1, +1#

#+1, +1, +1, color(red)(-1), color(red)(-1), color(red)(-1), +1#

#+1, +1, +1, color(red)(-1), color(red)(-1), +1, color(red)(-1)#

#+1, +1, +1, color(red)(-1), +1, color(red)(-1), +1#

#+1, +1, +1, color(red)(-1), +1, color(red)(-1), color(red)(-1)#

#+1, +1, +1, color(red)(-1), +1, +1, color(red)(-1)#

#+1, +1, +1, +1, color(red)(-1), +1, +1#

#+1, +1, +1, +1, color(red)(-1), color(red)(-1), +1#

#+1, +1, +1, +1, color(red)(-1), color(red)(-1), color(red)(-1)#

#+1, +1, +1, +1, color(red)(-1), +1, color(red)(-1)#

#+1, +1, +1, +1, +1, color(red)(-1), +1#

#+1, +1, +1, +1, +1, color(red)(-1), color(red)(-1)#

#+1, +1, +1, +1, +1, +1, color(red)(-1)#

I count #35# valid sequences.

Notes

Using this kind of approach we can manually or with computer assistance find the number of valid sequences for #n=1, 2, 3,...#

We can then look up the resulting sequence in the online encyclopedia of integer sequences https://oeis.org . They almost certainly have an entry for it with some interesting information.