A340352 Number of spanning trees of odd Aztec diamond OD_n.

1, 192, 4542720, 12116689944576, 3544863978266468352000, 112387469554685044937510092800000, 383669915612621265759587438135691539652804608, 140496256399491641572818822014023027580848616806252629983232

Offset

1,2

Comments

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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OD_1 OD_2 OD_3

Links

Table of n, a(n) for n=1..8.

D. E. Knuth, Aztec Diamonds, Checkerboard Graphs, and Spanning Trees, arXiv:math/9501234 [math.CO], 1995; J. Alg. Combinatorics 6 (1997), 253-257.

Formula

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).

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

Prog

(PARI) default(realprecision, 120);
{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)))}

Crossrefs

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

Sequence in context: A282541 A202930 A300848 * A012856 A114859 A298728

Adjacent sequences: A340349 A340350 A340351 * A340353 A340354 A340355

Keyword

nonn

Author

Seiichi Manyama, Jan 05 2021

Status

approved