Maat_Mons ninja'd me, but I know I can't go ahead and cancel this textwall. My heart is not strong enough to kill this baby. Ok I think a regular pattern can be guessed from that table (and the one on DMG too).
Assuming fifth increase of a 1d12 weapon is a typo (10d6 instead of 12d6), starting from 2d6 it goes like this:
Damage(0)= 2d6; dmg(1)=3d6; dmg(2)=4d6; dmg(3)=6d6...
Also assuming that Xd6/2 is equal to (X/2)d6 and different from Xd3
The succession is the sequent:
Damage(n)= 1d6 + [(Damage(n-1))/2] + [(Damage(n-2))/2]... + [(Damage(n-n) #or Damage(0)# /2]
Or, to put this EXTREMELY easier, start from 2d6->3d6 and double the number of d6 every two size increases. That's an approximation of multiplying by the square root of two every size increase, but the rounding prevents from making a formula from this consideration. Not really, but that would be a real mess (yeah... I'll probably won't sleep until I write it down on paper, I know...).
I'm sorry about the confusing parentheses. #text# is to separate the comment, obvious as it is, from the formula. Also[ x] is the integer part function.
This seems to work also on multiple d8, but there is only one given sample on the tables, as generally damage tends to be expressed in terms of d6.
Blame me if you math teacher comes in your nightmares tonight.