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Number of spanning trees of odd Aztec diamond OD_n.
2

%I #23 Jan 06 2021 04:23:53

%S 1,192,4542720,12116689944576,3544863978266468352000,

%T 112387469554685044937510092800000,

%U 383669915612621265759587438135691539652804608,140496256399491641572818822014023027580848616806252629983232

%N Number of spanning trees of odd Aztec diamond OD_n.

%C R. P. Stanley conjectured that the even Aztec diamond has exactly four times as many spanning trees as the odd Aztec diamond. This conjecture was first proved by D. E. Knuth.

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%C OD_1 OD_2 OD_3

%H D. E. Knuth, <a href="https://arxiv.org/abs/math/9501234">Aztec Diamonds, Checkerboard Graphs, and Spanning Trees</a>, arXiv:math/9501234 [math.CO], 1995; J. Alg. Combinatorics 6 (1997), 253-257.

%F a(n) = 4^(2*(n-1)) * A340166(n) = 4^(2*(n-1)*n) * Product_{1<=j,k<=n-1} (1 - sin(j*Pi/(2*n))^2 * sin(k*Pi/(2*n))^2).

%F a(n) ~ Gamma(1/4) * exp(8*G*n^2/Pi) / (Pi^(3/4) * sqrt(n) * 2^(2*n + 2)), where G is Catalan's constant A006752. - _Vaclav Kotesovec_, Jan 06 2021

%o (PARI) default(realprecision, 120);

%o {a(n) = round(4^(2*(n-1)*n)*prod(j=1, n-1, prod(k=1, n-1, 1-(sin(j*Pi/(2*n))*sin(k*Pi/(2*n)))^2)))}

%Y Cf. A007725 (even Aztec diamond), A340166, A340185 (halved Aztec diamond HOD_n).

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jan 05 2021