OFFSET
1,2
REFERENCES
Mihai Ciucu, Matchings and applications. Department of Mathematics, Indiana University, Bloomington, IN 47405. See Lecture 12.
FORMULA
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).
EXAMPLE
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, ...
PROG
(PARI) 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)))}
(Python)
# 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)])
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jan 07 2021
STATUS
approved