Recursive function that identifies whether string 1 is contained within string 2? (Python 3.4)

At the line:

 return contains_recursive(s1[0], s2[1:])

you are shortening s1 to one character but then at the next call you may hit:

 return contains_recursive(s1[1:], s2[1:])

with s1 a string of len 1.

You need to use:

return contains_recursive(s1, s2[1:])

and add a check on the length of s1

I take it that being recursive is a requirement. In that case:

def contains(s1, s2):
    if not s1:
        return True
    i = s2.find(s1[0])
    if i == -1:
        return False
    else:
        return contains(s1[1:], s2[i+1:])

This produces:

>>> contains("lit", "litter")
True
>>> contains("thot", "thurtle")
False
>>> contains("ratchet", "ramtbunchiousest")
True
>>> contains("shade", "hadsazie")
False

The error you are getting could be because s2 is an empty string. Add a check for it's length as well. If you reach this point it would mean you haven't found all the letters you are searching for and therefore the end result should be false.

if s2 == '':
    return False

Avoid using rescurive functions for efficiency.

def test(s1, s2):
    idx2 = 0
    for c in s1:
        if c in s2[idx2:]:
            idx2 = s2.index(c) + 1
        else:
            return False

    return True

# Test
lists = [   ("lit", "litter"), 
            ("thot", "thurtle"), 
            ("ratchet", "ramtbunchiousest"), 
            ("shade", "hadsazie")]

result = [test(*t) for t in lists]
print(result)
# Output
[True, False, True, False]