Consider triangular matrices $A = \left( {a(n,k)} \right)$ of arbitrary order with $a(n,k) = 0$ if $n + k$ is odd and $a(n,n - 2k) = \frac{{n!}}{{k!(n - 2k)!}}\frac{{(m + n - k - 1)!}}{{(m + n - 1)!}}$ for some positive integer $m.$

Let $B = \left( {b(n,k)} \right) = \log A$ be the triangular matrix satisfying $A = \exp B.$

Then calculations suggest that $$\frac{{b(n,n - 2k)}}{{a(n,n - 2k)}} = {A_k}\frac{{(m + n - 2k)!}}{{(m + n - k - 1)!}},$$ where the first terms of the sequence ${\left( {{A_k}} \right)_{k \ge 0}}$ are $${\left( {{A_k}} \right)_{k \ge 0}} = \left( {0,1, - 1,3, - 14,80, - 468,2268, \cdots } \right).$$

The same right-hand side appears for each positive integer $r$ if we choose $$a(n,n - 2k) = {r^k}\binom{\lfloor{{n}/2}\rfloor}k \prod\limits_{j = 0}^{k - 1} {\left( {\left\lfloor {\frac{{n + 1 - 2j}}{2}} \right\rfloor r - 1} \right)} \frac{{(m + n - k - 1)!}}{{(m + n - 1)!}}.$$

These $a(n,k)$ are the coefficients of orthogonal polynomials which I am studying.

The sequence ${\left( {{A_k}} \right)_{k \ge 0}}$ also appears in OEIS A027614 apparently in a similar setting.

By OEIS A179320 the exponential generating function $A(x) = \sum\limits_{n \ge 1} {\frac{{{A_n}}}{{n!}}{x^n}} $ satisfies $A(x) = \frac{{1 - x}}{{1 + x}}A\left( {\frac{x}{{{{(1 - x)}^2}}}} \right)$ and is uniquely determined by ${A_1} = 1$ and ${A_2} = - 1.$

I would be very interested to know if there are proofs known for some of these results and how to motivate the functional equation for the generating function of the coefficients of the matrix logarithms.