A007725 Number of spanning trees of Aztec diamonds of order n.
1, 4, 768, 18170880, 48466759778304, 14179455913065873408000, 449549878218740179750040371200000, 1534679662450485063038349752542766158611218432, 561985025597966566291275288056092110323394467225010519932928
Offset: 0
Keywords
Links
- 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
Programs
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Mathematica
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 *) -
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
Formula
a(n) ~ Gamma(1/4) * exp(8*G*n^2/Pi) / (Pi^(3/4) * sqrt(n) * 4^n), where G is 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
Extensions
More terms from Alois P. Heinz, Jan 20 2011
Offset changed (a(0)=1) by Seiichi Manyama, Jan 05 2021