What is the rule these numbers follow?

Question

Each row in this grid of numbers follows a specific pattern or rule:

 7   1   5   2  ||   8   4
12  10  14  13  ||  12   6
11   2  14   2  ||  15   6
 8   3  19   3  ||  21   6
 6   3   8   2  ||  16   3
10   3  19   3  ||  21   7
11   5  26   4  ||  28   9
 8   6  11   9  ||  22  19

There are two goals: Firstly, you must determine what the pattern is, and secondly, you must create a row of your own that follows the pattern.

Hint 1:

There is a reason that every second number in a row is smaller than the number that comes before it.

Hint 2:

The seperator is there for a purpose also

Hint 3:

This puzzle works regardless of base or representation

Hint 4 (major hint):

Half of the numbers, when removed can be uniquely replaced, while the other half each could potentially be replaced with one other number.

Answer 1

The pattern is that

If you treat pairs of numbers as binomial coefficients, multiplying the two coefficients on the left gives the value of the coefficient on the right.

Examples:

$$\binom{7}{1} \times \binom{5}{2} = \binom{8}{4}$$
$$\binom{12}{10} \times \binom{14}{13} = \binom{12}{6}$$
$$\binom{11}{2} \times \binom{14}{2} = \binom{15}{6}$$
$$\binom{6}{3} \times \binom{8}{2} = \binom{16}{3}$$
$$\binom{10}{3} \times \binom{19}{3} = \binom{21}{7}$$
$$\binom{11}{5} \times \binom{26}{4} = \binom{28}{9}$$
$$\binom{8}{6} \times \binom{11}{9} = \binom{22}{19}$$

Example of my own finding:

8 6 6 3 || 16 3