A340352 -id:A340352 - cp's OEIS frontend

A007725 Number of spanning trees of Aztec diamonds of order n.

1, 4, 768, 18170880, 48466759778304, 14179455913065873408000, 449549878218740179750040371200000, 1534679662450485063038349752542766158611218432, 561985025597966566291275288056092110323394467225010519932928

Offset: 0

Richard Stanley

nonn

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

Cf. A007726, A340166, A340176, A340185, A340352.

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 *)

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

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

More terms from Alois P. Heinz, Jan 20 2011

Offset changed (a(0)=1) by Seiichi Manyama, Jan 05 2021

A340425 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of spanning trees of odd Aztec rectangle of order (n, k).

Original entry on oeis.org

1, 4, 4, 16, 192, 16, 64, 8960, 8960, 64, 256, 417792, 4542720, 417792, 256, 1024, 19480576, 2280570880, 2280570880, 19480576, 1024, 4096, 908328960, 1143668117504, 12116689944576, 1143668117504, 908328960, 4096

Offset: 1

Seiichi Manyama, Jan 07 2021

			Square array begins:
    1,        4,            16,                64,                    256, ...
    4,      192,          8960,            417792,               19480576, ...
   16,     8960,       4542720,        2280570880,          1143668117504, ...
   64,   417792,    2280570880,    12116689944576,      64046643170770944, ...
  256, 19480576, 1143668117504, 64046643170770944, 3544863978266468352000, ...

Mihai Ciucu, Matchings and applications. Department of Mathematics, Indiana University, Bloomington, IN 47405. See Lecture 12.

Main diagonal gives A340352.

Cf. A340427.

default(realprecision, 120);
{T(n,k) = round(4^(2*n*k-n-k)*prod(a=1, n-1, prod(b=1, k-1, 1-(sin(a*Pi/(2*n))*sin(b*Pi/(2*k)))^2)))}

# Using graphillion
from graphillion import GraphSet
def make_OD_nk(n, k):
    n += 1
    k += 1
    grids = []
    s = k * n
    for i in range(1, k * n, k):
        for j in range(1, k):
            a, b = i + j - 1, i + j
            c = s + a
            if i > 1:
                grids.extend([(c - k, a), (c - k, b)])
            if i < k * (n - 1) + 1:
                grids.extend([(c, a), (c, b)])
    return grids
def A340425(n, k):
    universe = make_OD_nk(n, k)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
print([A340425(j + 1, i - j + 1) for i in range(7) for j in range(i + 1)])

T(n,k) = T(k,n).
T(n,k) = 4^(2*n*k-n-k) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(a*Pi/(2*n))^2 * sin(b*Pi/(2*k))^2).
T(n,k) = 4^(2*n*k-n-k) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - cos(a*Pi/(2*n))^2 * cos(b*Pi/(2*k))^2).
T(n,k) = 4^(2*n*k-n-k) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(a*Pi/(2*n))^2 * cos(b*Pi/(2*k))^2).

cp's OEIS Frontend

A007725 Number of spanning trees of Aztec diamonds of order n.

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