One more question related to my earlier "Special" meanders.

I am trying to isolate simplest problems related to it. Here is one.

For a composition (i. e. a tuple of natural numbers) $\boldsymbol a=(a_1,...,a_k)$, define the set of its midpoints by $$ \operatorname{mid}(\boldsymbol a):=\{a_1+...+a_{i-1}+\frac{a_i+1}2\mid1\leqslant i\leqslant k\}. $$ For example, $\operatorname{mid}(4,1,3)=\{\frac52,5,7\}$.

Call compositions $\boldsymbol a$ and $\boldsymbol b$ unmatchable if $\operatorname{mid}(\boldsymbol a)\cap\operatorname{mid}(\boldsymbol b)=\varnothing$.

I need any explicit information (formula, generating function, ...) for the numbers

$F(n):=$ number of unmatchable pairs $\langle\boldsymbol a,\boldsymbol b\rangle$ with $\sum a_i=\sum b_j=n$.

The sequence starts $0,2,6,24,78,284,960,3402,11710,41020,...$

My attempts so far have led to increasingly absurdly complicated approaches (like inverting infinite matrices with power series coefficients) and gave nothing in the end.

I believe a specialist can either give an answer immediately or relate it to some known difficult problem.

One more question related to my earlier "Special" meanders.

I am trying to isolate simplest problems related to it. Here is one.

For a composition (i. e. a tuple of natural numbers) $\boldsymbol a=(a_1,...,a_k)$, define the set of its midpoints by $$ \operatorname{mid}(\boldsymbol a):=\{a_1+...+a_{i-1}+\frac{a_i+1}2\mid1\leqslant i\leqslant k\}. $$ For example, $\operatorname{mid}(4,1,3)=\{\frac52,5,7\}$.

Call compositions $\boldsymbol a$ and $\boldsymbol b$ unmatchable if $\operatorname{mid}(\boldsymbol a)\cap\operatorname{mid}(\boldsymbol b)=\varnothing$.

I need any explicit information (formula, generating function, ...) for the numbers

$F(n):=$ number of unmatchable pairs $\langle\boldsymbol a,\boldsymbol b\rangle$ with $\sum a_i=\sum b_j=n$.

The sequence starts $0,2,6,24,78,284,960,3402,11710,41020,...$

My attempts so far have led to increasingly absurdly complicated approaches (like inverting infinite matrices with power series coefficients) and gave nothing in the end.

I believe a specialist can either give an answer immediately or relate it to some known difficult problem.