Comments on A181897 from Gregory Gerard Wojnar, Sep 29 2020

(1)
As the number of entries in row n of the triangle is the partition number p(n), cf. A000041, order all partitions of n as follows.  Let the first partition of n be the sum of n 1's [represented as (1,1,1,...,1)] and the p(n)th partition of n be simply n [represented as (n)]; representing each partition by listing its parts in increasing size [e.g., (1,1,2) represents the partition 1+1+2 of 4], order these representations in lexicographic order.  E.g., the 5 partitions of 4 are ordered (1,1,1,1) < (1,1,2) < (1,3) < (2,2) < (4). Sign each of the entries of the rows of this sequence with (-1)^k(P) where k(P) is the number of parts in partition P.  E.g., row 4 of this triangle becomes +1, -6, +8, +3, -6.  Then the sum of all such signed entries in each row is always 0, with the exception of row 1. 

(2)
Represent an integer partition of N as an N-tuple (m_j)_{j=1...N} such that Sum_{j=1...N}(j*m_j)=N, i.e., m_j is the multiplicity (i.e., the number of occurrences) of j in the partition.

For any such partition P, denote the number of parts of P by k(P)=Sum_{j=1...N}m_j.

For any such partition P, and for any square matrix A, define tr_P(A):= Product_{j=1...N}tr(A^j)^m_j where tr() denotes the trace of a square matrix.

For a fixed N, order the set of all integer partitions of N, represented as above, in reverse lexicographic order; equivalently, defining B^P:= Sum_{j=1...N}(m_j*B^(N-j) for any base B>= N+1, order the partitions from largest B^P value to smallest B^P value, with ord(P)=1 for the largest B^P and ord(P)=p(N) for the smallest B^P where p(N) equals the number of integer partitions of N, cf. A000041.

Let Part(N) denote the set of all integer partitions of N.

Finally, define for any nonnegative integer i, generalized determinants per D_i(A):= (1/i!)*Sum_{P in Part(i)}(-1)^(i-k(P))*T(n+1, ord(P)-1)*tr_P(A), where T(q, r) is this sequence, augmented by the additional datum T(0,1)=1 for a "row 0" not present in the above triangle [for the correspondence of the entries of row n of this triangle to partitions of n, this "row 0" corresponds to the trivial or empty partition of 0].

Then if A is an N X N matrix, it holds that D_N(A)=det(A) and the characteristic polynomial of A is c_A(t):= det(A-t.Id) = Sum_{i+j=N; 0<=i, j<=N}D_i(A)*(-t)^j where Id is the N X N identity matrix.  Thus the D_i(A) form a sequence of functions stepping from the trace tr(A) = D_1(A) to the determinant det(A) = D_N(A).

Further, these generalized determinants satisfy the recursion D_I(A) = (1/I)*Sum_{i=1...I; j=I-i}(-1)^i*tr(A^i)*D_j(A).