cp's OEIS Frontend

A340176 Number of spanning trees in the halved Aztec diamond HMD_n.

%I A340176 #67 Feb 28 2023 23:47:25
%S A340176 1,1,4,208,121856,772189440,51989627289600,36837279603595907072,
%T A340176 273129993621426778551615488,21114078836429317912110529666154496,
%U A340176 16975032309392309949804839529585109326888960
%N A340176 Number of spanning trees in the halved Aztec diamond HMD_n.
%C A340176                                   *---*
%C A340176                                   |   |
%C A340176               *---*           *---*---*---*
%C A340176               |   |           |   |   |   |
%C A340176   *---*   *---*---*---*   *---*---*---*---*---*
%C A340176   HMD_1       HMD_2               HMD_3
%C A340176 -------------------------------------------------
%C A340176               *---*
%C A340176               |   |
%C A340176           *---*---*---*
%C A340176           |   |   |   |
%C A340176       *---*---*---*---*---*
%C A340176       |   |   |   |   |   |
%C A340176   *---*---*---*---*---*---*---*
%C A340176               HMD_4
%H A340176 Seiichi Manyama, <a href="/A340176/b340176.txt">Table of n, a(n) for n = 0..45</a>
%H A340176 Mihai Ciucu, <a href="https://arxiv.org/abs/0710.4500">Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole</a>, arXiv:0710.4500 [math.CO], 2007. See Corollary 3.6.
%H A340176 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>
%H A340176 Wikipedia, <a href="https://en.wikipedia.org/wiki/Resultant">Resultant</a>
%F A340176 a(n) = Product_{1<=j<k<=2*n-1 and j+k<=2*n-1} (4 - 4*cos(j*Pi/(2*n))*cos(k*Pi/(2*n))).
%F A340176 a(n) = 2^(n-1) * A007726(n) * A334089(n) = sqrt(A007341(n) * A334088(n) / n) for n > 0.
%F A340176 a(n) = 4^(n-1) * A340139(n) = 4^((n-1)^2) * Product_{1<=j<k<=n-1} (1 - cos(j*Pi/(2*n))^2 * cos(k*Pi/(2*n))^2) for n > 0. - _Seiichi Manyama_, Jan 02 2021
%F A340176 a(n) ~ sqrt(Gamma(1/4)) * exp(4*G*n^2/Pi) / (Pi^(3/8) * n^(3/4) * 2^(n - 1/4) * (1 + sqrt(2))^n), where G is Catalan's constant A006752. - _Vaclav Kotesovec_, Jan 05 2021
%e A340176 a(2) = 4;
%e A340176       *   *           *---*           *---*           *---*
%e A340176       |   |               |           |               |   |
%e A340176   *---*---*---*   *---*---*---*   *---*---*---*   *---*   *---*
%o A340176 (PARI) default(realprecision, 120);
%o A340176 {a(n) = round(prod(j=1, 2*n-1, prod(k=j+1, 2*n-1-j, 4-4*cos(j*Pi/(2*n))*cos(k*Pi/(2*n)))))}
%o A340176 (PARI) {a007341(n) = polresultant(polchebyshev(n-1, 2, x/2), polchebyshev(n-1, 2, (4-x)/2))};
%o A340176 {a334088(n) = sqrtint(polresultant(polchebyshev(2*n, 1, x/2), polchebyshev(2*n, 1, I*x/2)))};
%o A340176 {a(n) = if(n==0, 1, sqrtint(a007341(n)*a334088(n)/n))}
%o A340176 (PARI) default(realprecision, 120);
%o A340176 {a(n) = if(n==0, 1, round(4^((n-1)^2)*prod(j=1, n-1, prod(k=j+1, n-1, 1-(cos(j*Pi/(2*n))*cos(k*Pi/(2*n)))^2))))} \\ _Seiichi Manyama_, Jan 02 2021
%o A340176 (Python)
%o A340176 # Using graphillion
%o A340176 from graphillion import GraphSet
%o A340176 def make_HMD(n):
%o A340176     s = 1
%o A340176     grids = []
%o A340176     for i in range(2 * n, 0, -2):
%o A340176         for j in range(i - 2):
%o A340176             a, b, c = s + j, s + j + 1, s + i + j
%o A340176             grids.extend([(a, b), (b, c)])
%o A340176         grids.append((s + i - 2, s + i - 1))
%o A340176         s += i
%o A340176     return grids
%o A340176 def A340176(n):
%o A340176     if n == 0: return 1
%o A340176     universe = make_HMD(n)
%o A340176     GraphSet.set_universe(universe)
%o A340176     spanning_trees = GraphSet.trees(is_spanning=True)
%o A340176     return spanning_trees.len()
%o A340176 print([A340176(n) for n in range(7)])
%Y A340176 Cf. A007341, A007725, A007726, A334088, A334089, A340139, A340166, A340185 (halved Aztec diamond HOD_n).
%K A340176 nonn
%O A340176 0,3
%A A340176 _Seiichi Manyama_, Dec 31 2020