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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*---*---* *---*---*---*---* *---*---*---*---*---*---*
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* *---*---* *---*---*---*---*
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*
OD_1 OD_2 OD_3
LINKS
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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 05 2021
STATUS
approved