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A367252 a(n) is the number of ways to tile an n X n square as explained in comments.

%I A367252 #25 Nov 14 2023 11:57:37
%S A367252 1,0,1,4,88,3939,534560,185986304,175655853776,437789918351688,
%T A367252 2898697572048432368,50698981110982431863735,
%U A367252 2342038257118692026082013568,285250169294740386915765591840768,91531011920509198679773321121428857296,77312253225939431362091700178995800855209496
%N A367252 a(n) is the number of ways to tile an n X n square as explained in comments.
%C A367252 Draw a Dyck path from (0,0) to (n,n) so the path always stays above the diagonal. Now section the square into horizontal rows of height one to the left of the path and tile these rows using 1 X 2 and 1 X 1 tiles.  Similarly, section the part to the right of the path into columns with width one and tile these using 2 X 1 and 1 X 1 tiles. Furthermore, no 1 X 1 tiles are allowed in the bottom row.
%H A367252 Anna Tscharre, <a href="/A367252/a367252.pdf">A possible 6 X 6 tiling</a>
%F A367252 a(n) == 1 (mod 2) <=> n in { A055010 }. - _Alois P. Heinz_, Nov 11 2023
%p A367252 b:= proc(x, y) option remember; (F->
%p A367252      `if`(x=0 and y=0, 1, `if`(x>0, b(x-1, y)*F(y-1), 0)+
%p A367252      `if`(y>x, b(x, y-1)*F(x+1), 0)))(combinat[fibonacci])
%p A367252     end:
%p A367252 a:= n-> b(n$2):
%p A367252 seq(a(n), n=0..15);  # _Alois P. Heinz_, Nov 11 2023
%t A367252 b[x_, y_] := b[x, y] = With[{F = Fibonacci},
%t A367252      If[x == 0 && y == 0, 1,
%t A367252      If[x > 0, b[x - 1, y]*F[y - 1], 0] +
%t A367252      If[y > x, b[x, y - 1]*F[x + 1], 0]]];
%t A367252 a[n_] := b[n, n];
%t A367252 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Nov 14 2023, after _Alois P. Heinz_ *)
%Y A367252 Special case of A003150.
%Y A367252 Cf. A055010, A307082.
%K A367252 nonn
%O A367252 0,4
%A A367252 _Anna Tscharre_, Nov 11 2023