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By construction the first column must be the alternating sign prime number sequence such that when $z=0$, the equation reduces to the primes. We can attempt to find the sequences however for the diagonals of this table! The main diagonal appears to be negative the sequence OEIS A007504, which is the sum of the first few primes. This is achieved also by extending $table 1$ back by one extra row.  Looking a the next diagonal down, this appears to be the partial sum of the sequence OEIS A117495, which is the product of a prime with the number of primes less than itself. That is $p_n\pi(p_n-1)$ when $pi(x)$ $\pi(x)$  is defined as the number of primes smaller or equal to $x$.