Find an expression for the sum of the i-th row of the following triangle, and prove its correctness. Each entry is the sum of the three entries directly above it. All non-existing entries are considered 0.

(note its in a pyramid like appearance but cheg edited it to be like a right triangle)

1

1 1 1

1 2 3 2 1

1 3 6 7 6 3 1

1 4 10 16 19 16 10 4 1