Say you're merging two nodes a and b, into res.

res.triplets = a.triplets + b.triplets + (a.pairs * b.sum) + (a.sum * b.pairs)

res.pairs = a.pairs + b.pairs + a.sum * b.sum

res.sum = a.sum + b.sum

Initialize single element nodes with (0, 0, arr[i]).

I'm not sure how to explain the general process, but try defining what you're trying to calculate into simpler things, even if it means defining new things to store. For e.g. in triplets, either they'll be completely on left or right side, or 2 will be on one side, one on the other. How do you get product of 2 at a time? They can be both on left, both right, or one each side. Keep doing until you have merging logic for all things you store in your segment tree nodes.

Do these for a bit of practice:

https://www.spoj.com/problems/GSS1/ https://www.spoj.com/problems/GSS3/ https://www.spoj.com/problems/GSS5/ https://www.codechef.com/problems/HARRYGOF