What you could do is say the kids got a, b, c, d, d (the twins each got the same) and then seek quadruples satisfiing a+b+c+2d=13 than try each value of d.

d=0: 13 candies 3 kids place dividers in ccccccccccccc (c for CANDY!) and see the number of ways we can put the 3 dividers (3 kids left) so 15 choose 2 = 105

d=1: same, 13 choose 2 = 78

d=2: same logic, 11 choose 2 = 55

d=3: ditto I stop saying same 9 choose 2 = 36

d=4: 7 choose 2 = 21

d=5: 5 choose 2 = 10

d=6: 3 choose 2 = 3

105+78+55+36+21+10+3=308

I got 308