A007726 Number of spanning trees of quarter Aztec diamonds of order n.

1, 1, 4, 56, 2640, 411840, 210613312, 351102230528, 1901049105201408, 33349238079515381760, 1892086487183556298556416, 346728396311328694807284940800, 205021218459835103075295973360128000, 390870571052378289975757743555515137130496

Offset: 1

Author

Richard Stanley

Keywords

• nonn

References

• Mihai Ciucu (ciucu(AT)math.gatech.edu), in preparation, 2001.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..50
• 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.
• R. Kenyon, J. Propp and D. Wilson, Trees and matchings, Electronic Journal of Combinatorics, 7(1):R25, 2000.
• 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

Crossrefs

Cf. A007725, A007341, A065072, A340052.

Programs

Mathematica

Table[Product[Product[4 - 2*Cos[j*Pi/n] - 2*Cos[k*Pi/n], {j, 1, k-1}], {k, 2, n-1}], {n, 1, 15}] // Round (* Vaclav Kotesovec, Dec 30 2020 *)
Table[Sqrt[Resultant[ChebyshevU[n-1, x/2], ChebyshevU[n-1, (4-x)/2], x] / (n * 2^(n-1))], {n, 1, 15}] (* Vaclav Kotesovec, Dec 30 2020 *)

PARI

default(realprecision, 120);
{a(n) = round(prod(j=2, n-1, prod(i=1, j-1, 4*sin(i*Pi/(2*n))^2+4*sin(j*Pi/(2*n))^2)))} \\ Seiichi Manyama, Dec 29 2020

Formula

a(n) = Product_{0Sean A. Irvine, Jan 20 2018

From Vaclav Kotesovec, Dec 30 2020: (Start)

a(n) ~ sqrt(Gamma(1/4)) * 2^(5/8) * exp(2*G*n^2/Pi) / (Pi^(3/8) * n^(3/4) * 2^(n/2) * (1 + sqrt(2))^n), where G is Catalan's constant A006752.

a(n) = sqrt(A007341(n) / (n * 2^(n-1))). (End)

Extensions

More terms from Sean A. Irvine, Jan 20 2018