Dear Doug,

Here is, in more detail, the evaluation of det(-I+M).

  With best wishes,

   Christian

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\def\KratBH{3}
\def\AndrAS{1}
\def\MiRRAD{4}
\def\KratBD{2}

Similar to \cite{\MiRRAD}, define determinants $Z_n(x,\mu)$ by
$$Z_n(x,\mu)=\det_{0\le i,j\le n-1}\left(-\delta_{ij}+\sum_{t,k=0}
^{n-1}\binom {i+\mu}t\binom kt\binom
{j-k+\mu-1}{j-k}x^{k-t}\right).\tag1$$
The only difference to the definition of $Z_n(x,\mu)$ in
\cite{\MiRRAD} is the minus sign in front of $\delta_{ij}$.

Then an analogue of Theorem~5 of \cite{\MiRRAD} is true.
\proclaim{Lemma}Let $n$ be a nonegative integer. Then $Z_n(x,\mu)=0$
if $n$ is odd. If $n$ is even, then $Z_n(x,\mu)$ factors,
$$\multline
Z_n(x,\mu)=(-1)^{n/2}\det_{0\le i,j\le n/2-1}\left(\sum _{t=0} ^{n-1}\frac {t+1}
{j+1}\binom {i+\mu}{t-i}\binom {j+1}{t-j}x^{2j+1-t}\right)\\
\times
\det_{0\le i,j\le n/2-1}\left(\sum _{t=0} ^{n-1}\frac {t+\mu+1}
{i+\mu+1}\binom {i+\mu+1}{t-i}\binom {j}{t-j}x^{2j-t}\right).
\endmultline\tag2$$
\endproclaim
\demo{Proof}Define matrices $S,M,U$ as in the proof of Theorem~5 of
\cite{\MiRRAD} and do the manipulations given there, but starting
with $-I+SMU$. The result is that 
$$Z_n(x,\mu)=\det_{0\le i,j\le n-1}(-I+SMU)=\det_{0\le i,j\le n-1}(-V+W),$$
where
$$\align V_{2i+r,2j+s}&=\sum _{t=0} ^{n-1}(-1)^{r+s}\binom
{i+r+\mu}{t-i}\binom {j+s}{t-j}x^{2j+s-t},\\
W_{2i+r,2j+s}&=\sum _{t=0} ^{n-1}(-1)^{r+s}\binom
{i+\mu}{t-i-r}\binom {j}{t-j-s}x^{2j+s-t},\endalign$$
where $r$ and $s$ are restricted to be $0$ or $1$, as in
\cite{\MiRRAD}.

It is straightforward to check that $V_{2i,2j}=W_{2i,2j}$. Hence,
each entry of the
matrix $-V+W$ in an even numbered row and even numbered column is 0.
This implies that $\det(-V+W)$ must be 0 whenever the size of the
determinant, $n$, is odd. In the case that $n$ is even it implies
the factorization
$$\multline
Z_n(x,\mu)=\det(-V+W)\\
=(-1)^{n/2}\det_{0\le i,j\le n/2-1}(-V_{2i,2j+1}+W_{2i,2j+1})
\det_{0\le i,j\le n/2-1}(-V_{2i+1,2j}+W_{2i+1,2j}).
\endmultline$$
As is easily verified, this equation is exactly equivalent to (2).\quad \quad
\qed
\enddemo

Now choose $x=1$ in the Lemma. Then all the sums appearing in (2) can be
evaluated by means of the Vandermonde summation. The result is
$$\multline
Z_n(1,\mu)=\det_{0\le i,j\le n-1}\left(-\delta_{ij}+\binom {i+j+2\mu}{j}\right)\\
=\det_{0\le i,j\le n/2-1}\left((3i+2\mu+1)
\frac {(i+j+\mu)!}{(2i-j+\mu)!\,(2j-i+1)!}\right)\\
\times
\det_{0\le i,j\le n/2-1}\left((3j+\mu+1)
\frac {(i+j+\mu)!}{(2i-j+\mu+1)!\,(2j-i)!}\right).
\endmultline\tag3$$
Both determinants on the right-hand side of this identity can be
evaluated by means of Theorem~10 in \cite{\KratBD}, which reads
$$\multline \det_{0\le i,j\le m-1}\left(\frac {(x+y+i+j-1)!}
{(x+2i-j)!\,(y+2j-i)!}\right)\\
=\prod _{i=0} ^{m-1}\frac {i!\,(x+y+i-1)!\,(2x+y+2i)_i\,(x+2y+2i)_i}
{(x+2i)!\,(y+2i)!}.
\endmultline\tag4$$

In summary, we have proved the following theorem.
\proclaim{Theorem}For nonnegative integers $n$ we have
$$\multline 
\det_{0\le i,j\le n-1}\left(-\delta_{ij}+\binom {i+j+2\mu}{j}\right)
\\=
\cases 0,&\text {if $n$ is odd,}\\
(-1)^{n/2}\prod _{i=0} ^{n/2-1}\frac {i!^2\,(\mu+i)!^2\,
(3i+\mu+1)!^2\,(3i+2\mu+1)!^2} {(2i)!\,(2i+1)!\,(\mu+2i)!^2\,
(\mu+2i+1)!^2\,(2\mu+2i)!\,(2\mu+2i+1)!},
&\text {if $n$ is even.}\endcases
\endmultline$$
\endproclaim

In analogy with what is going on for the Andrews determinant (i.e.,
the determinant (1) with $x=1$ and 
with $-\delta_{ij}$ replaced by $\delta_{ij}$), I
was tempted to see if the above determinants factor for
$x=2$ as well. Of course, they do. However, as far as I can see, this
does not follow from any of the known results. The relevant
references for the factorization of the ``usual" $Z_n(2,\mu)$ are
\cite{\AndrAS} and \cite{\KratBH}.


\Refs

\ref\no \AndrAS\by G. E. Andrews \yr 1987 
\paper Plane partitions (IV): A conjecture of Mills--Robbins--Rumsey
\jour Aequationes Math\.\vol 33
\pages 230--250\endref

\ref\no \KratBD\by C.    Krattenthaler \yr 1997 
\paper Determinant identities and a generalization of the number of totally symmetric self-complementary plane partitions
\jour Electron\. J. Combin\.\vol 4\rm(1)
\pages \#R27, 62~pp\endref

\ref\no \KratBH\by C.    Krattenthaler 
\paper A new proof of the M--R--R conjecture --- including a generalization
\jour J. Difference Equ\. Appl\. \toappear\vol 
\pages \endref

\ref\no \MiRRAD\by W. H. Mills, D. H. Robbins and H. Rumsey \yr 1987 
\paper Enumeration of a symmetry class of plane partitions
\jour Discrete Math.\vol 67
\pages 43--55 \endref

\endRefs
\end
