Non-Associate Powers and a Functional Equation

Several writers have studied algebras in which multiplication is non-associative, that is, x yz ≠ xy z . It is necessary in a non-associative algebra to distinguish the possible interpretations of a power x n In a non-commutative non-associative algebra x 2 is unique, x 3 can mean xx 2 or x 2 x ; x 4 can mean x xx 2 , x x 2 x , x 2 x 2 , xx 2 x or x 2 x x , x 5 has 14 interpretations; x 6 has 42; and so on. In a commutative non-associative algebra, the possible interpretations are fewer x 3 is unique, x 4 can mean xx 3 or x 2 x 2 , x 5 can mean x xx 3 , x x 2 x 2 or x 2 x 3 , x 6 has 6 interpretations, and so on. The problem considered here is how many meanings are there for x n (A) in a general non-commutative non-associative algebra ? (B) in a general commutative non-associative algebra ? The answer to (A) is I am not able to find any such simple formula for (B).