TITLE: in search of a transformation between determinants
QUESTION [13 upvotes]: Motivated by this MO question. Consider the two matrices $A_n$ and $B_n$ with entries $\binom{2j}i$ and $\binom{n+1}{2j-i}$, respectively; for $1\leq i, \,j\leq n$.
I can show $\det A_n=\det B_n=2^{\binom{n+1}2}$; it is not a problem (see, for instance, my answer here).

Question.
(1) Is there a transformation converting $A_n$ into $B_n$ (or vice-versa)?
(2) Is there a combinatorial interpretation/bijection revealing the counts by $\det A_n$ and $\det B_n$?

My asking for (2) is due to the fact that $2^{\binom{n+1}2}$ enumerates domino tilings of an Aztec diamond.

(3) Let $s(k)=$the number of $1$'s in the binary expansion of $k$. Then, $A_n$ and $B_n$ share the same Smith normal form (showing the diagonal vector) given by
$$[2^{\max(4k-2n+s(n-k)-s(k),0)}:\, 1\leq k\leq n].$$
This claim is based on data from Noam Elkies' comments seen below. Any proof?

Remark. Let $\lceil x\rceil=$the smallest integer greater than or equal to $x$ (ceiling function). So, (3) implies
$$\sum_{k=1}^n\max(4k-2n+s(n-k)-s(k),0)=\frac{n(n+1)}2;$$
Or, equivalently $\sum_{k=\lceil\frac{n}2\rceil}^n(s(k)-s(n-k))=\lceil\frac{n}2\rceil$.

REPLY [9 votes]: There is such a transformation, of the form predicted in Linear transformation that preserves the determinant.
Denoting $R$  the involution matrix $e_i\mapsto e_{n+1-i}$, it turns out that the matrix $A$ has an $LU$-decomposition in which $U:=\left[{j\choose i}\right]_{{1\le i\le n}\atop{1\le j\le n}}$, and the lower triangular part  is  $L=RCR$, where $C$ is the upper triangular matrix in Suvrit's decomposition , 
$B=VCV^{-1}$ (warning:  $B$ of this question is named "$A$" there). So $B= (VR) A(VRU)^{-1}$, with $\det(VR)=\det(VRU)^{-1}=1$.
$$*$$
[edit] The description becomes a bit more gracious if we include the indices $i=0$ and $j=0$. So, if we define the $n\times n$ matrices with integer entries
$$A_n:=\left[{2j\choose i}\right]_{{0\le i< n}\atop{0\le j< n}}\qquad B_n:=\left[{n\choose 2j-i}\right]_{{0\le i< n}\atop{0\le j< n}}$$
$$U_n:=\left[{j\choose i}\right]_{{0\le i< n}\atop{0\le j< n}}\qquad L_n:=\left[2^{2j-i}{j\choose2j- i}\right]_{{0\le i< n}\atop{0\le j< n}}$$
$$N_n:=\Big[ \delta_{i+1,j}\Big]_{{0\le i< n}\atop{0\le j< n}}\qquad R_n:=\Big[ \delta_{n-i,j}\Big]_{{0\le i< n}\atop{0\le j< n}}$$
Then, (hiding the subscript $n$) 
$$A=LU$$
and
$$B=VCV^{-1}$$
with
$$V:=U^{T}R\qquad C:=(I+N)AU^{-1}\ .$$