This is ordinal notation i came up recently.
NOTE: This is not the true O ordinal notation. Original O ordinal notation will come soon.
Basic (Extension 0)
O(0) = 0
2-entry array (Extension 1)
O(α,β) = A(α,β)
Linear array (Extension 2)
O(α,...,β) = φ(α,...,β)
Nested Linear Array (Extension 3)
& = any array O(α(β)) = O(α,α,α,...,α,α,α) with β entries
O(α(&)) = the same as O(&) but it modifies the () instead of the main function
Hypernested Linear Array (Extension 4)
Arrow Nested Linear Array (Extension 5)
More extensions coming soon!