Suppose I have an nxn square table; and I want to create a list that
will tell me, for the jth row, the sum of the first j columns.

Is there any way to enter the formula so that, say, when I drag down to
create a column of values, I don't have to go through and manually edit
each one?

Here's one way.

With the table starting in A1, and with the size (your "n") in N1, put this
=SUM(OFFSET($A$1,0,0,$N$1,ROW()))
in row 1 of some column and drag it down. Then the second cell down has the sum of the first two columns of the table; the third cell, three columns; etc.

Explanation: OFFSET() returns an array, and can be used as a function argument where an array is expected. The arguments of OFFSET() allow you to vary the placement of the array and its height and width.

Modify and expand as needed.