python sum function - `start` parameter explanation required

Sum does something like this

def sum(values, start = 0):
    total = start
    for value in values:
        total = total + value
    return total

sum([1,2],[3,4]) expands something like [3,4] + 1 + 2, which you can see tries to add numbers and lists together.

In order to use sum to produce lists, the values should be a list of lists, whereas start can be just a list. You'll see in your failing examples that the list contains at least some ints, rather then all lists.

The usual case where you might think of using sum with lists is to convert a list of lists into a list

sum([[1,2],[3,4]], []) == [1,2,3,4]

But really you shouldn't do that, as it'll be slow.

a=[[1, 20], [2, 3]]
b=[[[[[[1], 2], 3], 4], 5], 6]
sum(b, a)

This error has nothing to do with the start parameter. There are two items in the list b. One of them is [[[[[1], 2], 3], 4], 5], the other is 6, and a list and int cannot be added together.

sum(a, b)

This is adding:

[[[[[[1], 2], 3], 4], 5], 6] + [1, 20] + [2, 3]

Which works fine (as you're just adding lists to lists).

a=[1,2]
b=[3,4]
sum(a,b)

This is trying to add [3,4] + 1 + 2, which again isn't possible. Similarly, sum(b,a) is adding [1, 2] + 3 + 4.

What if the start is not a string and not an integer?

sum can't sum strings. See:

>>> sum(["a", "b"], "c")
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
TypeError: sum() can't sum strings [use ''.join(seq) instead]

One of the things that has been hinted at but not explicitly stated in the other answers is that the start value defines the type for the return value and for the items being summed. Because the default is start=0, (and 0 is an integer, of course) all items in the iterable must be integers (or types with an __add__ method that works with integers). Other examples have mentioned concatenating lists:

(sum([[1,2],[3,4]], []) == [1,2,3,4])

or timedate.timedelta objects:

(sum([timedelta(1), timedelta(2)], timedelta()) == timedelta(3)).

Notice that both examples pass an empty object of the type in the iterable as the start parameter to avoid getting a TypeError: unsupported operand type(s) for +: 'int' and 'list' error.