The New Notation for Hyperoperation of a Sequence

For a sequence \(a_1, a_2, \ldots, a_n\), we define the exponent, tetration, and pentation of a sequence \(a_n\) as \(\overset{n}{\underset{k = 1}{\textrm{E}}} (a_k) = a_1[3]a_2[3]\cdots[3]a_n\), \(\overset{n}{\underset{k = 1}{\textrm{T}}} (a_k) = a_1[4]a_2[4]\cdots[4]a_n\), \(\overset{n}{\underset{k = 1}{\mathrm{\Phi}}} (a_k) = a_1[5]a_2[5]\cdots[5]a_n\). Also, we define the \(i\)-th hyperoperation of a sequence \(a_n\) as \(\overset{n}{\underset{k = 1}{\textrm{H}_i}} (a_k) = a_1[i]a_2[i]\cdots[i]a_n\).