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A340425 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of spanning trees of odd Aztec rectangle of order (n, k).

%I A340425 #22 Jan 07 2021 19:25:23
%S A340425 1,4,4,16,192,16,64,8960,8960,64,256,417792,4542720,417792,256,1024,
%T A340425 19480576,2280570880,2280570880,19480576,1024,4096,908328960,
%U A340425 1143668117504,12116689944576,1143668117504,908328960,4096
%N A340425 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of spanning trees of odd Aztec rectangle of order (n, k).
%D A340425 Mihai Ciucu, Matchings and applications. Department of Mathematics, Indiana University, Bloomington, IN 47405. See Lecture 12.
%F A340425 T(n,k) = T(k,n).
%F A340425 T(n,k) = 4^(2*n*k-n-k) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(a*Pi/(2*n))^2 * sin(b*Pi/(2*k))^2).
%F A340425 T(n,k) = 4^(2*n*k-n-k) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - cos(a*Pi/(2*n))^2 * cos(b*Pi/(2*k))^2).
%F A340425 T(n,k) = 4^(2*n*k-n-k) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(a*Pi/(2*n))^2 * cos(b*Pi/(2*k))^2).
%e A340425 Square array begins:
%e A340425     1,        4,            16,                64,                    256, ...
%e A340425     4,      192,          8960,            417792,               19480576, ...
%e A340425    16,     8960,       4542720,        2280570880,          1143668117504, ...
%e A340425    64,   417792,    2280570880,    12116689944576,      64046643170770944, ...
%e A340425   256, 19480576, 1143668117504, 64046643170770944, 3544863978266468352000, ...
%o A340425 (PARI) default(realprecision, 120);
%o A340425 {T(n,k) = round(4^(2*n*k-n-k)*prod(a=1, n-1, prod(b=1, k-1, 1-(sin(a*Pi/(2*n))*sin(b*Pi/(2*k)))^2)))}
%o A340425 (Python)
%o A340425 # Using graphillion
%o A340425 from graphillion import GraphSet
%o A340425 def make_OD_nk(n, k):
%o A340425     n += 1
%o A340425     k += 1
%o A340425     grids = []
%o A340425     s = k * n
%o A340425     for i in range(1, k * n, k):
%o A340425         for j in range(1, k):
%o A340425             a, b = i + j - 1, i + j
%o A340425             c = s + a
%o A340425             if i > 1:
%o A340425                 grids.extend([(c - k, a), (c - k, b)])
%o A340425             if i < k * (n - 1) + 1:
%o A340425                 grids.extend([(c, a), (c, b)])
%o A340425     return grids
%o A340425 def A340425(n, k):
%o A340425     universe = make_OD_nk(n, k)
%o A340425     GraphSet.set_universe(universe)
%o A340425     spanning_trees = GraphSet.trees(is_spanning=True)
%o A340425     return spanning_trees.len()
%o A340425 print([A340425(j + 1, i - j + 1) for i in range(7) for j in range(i + 1)])
%Y A340425 Main diagonal gives A340352.
%Y A340425 Cf. A340427.
%K A340425 nonn,tabl
%O A340425 1,2
%A A340425 _Seiichi Manyama_, Jan 07 2021