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A007725
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Number of spanning trees of Aztec diamonds of order n.
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7
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1, 4, 768, 18170880, 48466759778304, 14179455913065873408000, 449549878218740179750040371200000, 1534679662450485063038349752542766158611218432, 561985025597966566291275288056092110323394467225010519932928
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OFFSET
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0,2
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LINKS
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Table of n, a(n) for n=0..8.
Timothy Y. Chow, The Q-spectrum and spanning trees of tensor products of bipartite graphs, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3155-3161.
D. E. Knuth, Aztec Diamonds, Checkerboard Graphs, and Spanning Trees, arXiv:math/9501234 [math.CO], 1995; J. Alg. Combinatorics 6 (1997), 253-257.
R. P. Stanley, Spanning trees of Aztec diamonds, Discrete Math. 157 (1996), 375-388 (Problem 251).
Index entries for sequences related to trees
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FORMULA
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a(n) ~ Gamma(1/4) * exp(8*G*n^2/Pi) / (Pi^(3/4) * sqrt(n) * 4^n), where G is the Catalan's constant A006752. - Vaclav Kotesovec, Jan 05 2021
a(n) = 4^(2*n-1) * Product_{1<=j,k<=n-1} (4 - 4*cos(j*Pi/(2*n))*cos(k*Pi/(2*n)))*(4 + 4*cos(j*Pi/(2*n))*cos(k*Pi/(2*n))); [Knuth Eq. (8) p. 3]. - Seiichi Manyama, Jan 05 2021
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MATHEMATICA
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Table[4^n * Product[Product[4 - 4*Cos[j*Pi/(2*n)]*Cos[k*Pi/(2*n)], {k, 1, n-1}], {j, 1, 2*n-1}], {n, 0, 10}] // Round (* Vaclav Kotesovec, Jan 05 2021 *)
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PROG
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(PARI) default(realprecision, 120);
{a(n) = if(n==0, 1, round(4^(2*(n-1)*n+1)*prod(j=1, n-1, prod(k=1, n-1, 1-(sin(j*Pi/(2*n))*sin(k*Pi/(2*n)))^2))))} \\ Seiichi Manyama, Jan 05 2021
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CROSSREFS
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Cf. A007726, A340166, A340176, A340185, A340352.
Sequence in context: A284813 A306254 A068112 * A102195 A114766 A239623
Adjacent sequences: A007722 A007723 A007724 * A007726 A007727 A007728
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KEYWORD
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nonn
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AUTHOR
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Richard Stanley
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EXTENSIONS
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More terms from Alois P. Heinz, Jan 20 2011
Offset changed (a(0)=1) by Seiichi Manyama, Jan 05 2021
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STATUS
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approved
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