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A340185 Number of spanning trees in the halved Aztec diamond HOD_n. 6
1, 1, 15, 2639, 5100561, 105518291153, 23067254643457375, 52901008815129395889375, 1266973371422697144030728637409, 315937379766837559600972497421046382689, 818563964325891485548944567913851815851212484079 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

                                              *

                                              |

                      *                   *---*---*

                      |                   |   |   |

      *           *---*---*           *---*---*---*---*

      |           |   |   |           |   |   |   |   |

  *---*---*   *---*---*---*---*   *---*---*---*---*---*---*

    HOD_1           HOD_2                   HOD_3

-------------------------------------------------------------

                  *

                  |

              *---*---*

              |   |   |

          *---*---*---*---*

          |   |   |   |   |

      *---*---*---*---*---*---*

      |   |   |   |   |   |   |

  *---*---*---*---*---*---*---*---*

                HOD_4

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..40

Mihai Ciucu, Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole, arXiv:0710.4500 [math.CO], 2007. See Corollary 3.7.

FORMULA

a(n) = Product_{1<=j<k<=2*n and j+k<=2*n} (4 - 4*cos(j*Pi/(2*n+1))*cos(k*Pi/(2*n+1))).

From Seiichi Manyama, Jan 02 2021: (Start)

a(n) = 4^((n-1)*n) * Product_{1<=j<k<=n} (1 - cos(j*Pi/(2*n+1))^2 * cos(k*Pi/(2*n+1))^2).

a(n) = A340052(n) * A065072(n) = (1/2^n) * sqrt(A127605(n) * A004003(n) / (2*n+1)). (End)

a(n) ~ sqrt(Gamma(1/4)) * exp(G*(2*n+1)^2/Pi) / (Pi^(3/8) * n^(3/4) * 2^(n + 3/4) * (1 + sqrt(2))^(n + 1/2)), where G is the Catalan's constant A006752. - Vaclav Kotesovec, Jan 03 2021

MATHEMATICA

Table[4^((n-1)*n) * Product[Product[(1 - Cos[j*Pi/(2*n + 1)]^2*Cos[k*Pi/(2*n + 1)]^2), {k, j+1, n}], {j, 1, n}], {n, 0, 12}] // Round (* Vaclav Kotesovec, Jan 03 2021 *)

PROG

(PARI) default(realprecision, 120);

{a(n) = round(prod(j=1, 2*n, prod(k=j+1, 2*n-j, 4-4*cos(j*Pi/(2*n+1))*cos(k*Pi/(2*n+1)))))}

(PARI) default(realprecision, 120);

{a(n) = round(4^((n-1)*n)*prod(j=1, n, prod(k=j+1, n, 1-(cos(j*Pi/(2*n+1))*cos(k*Pi/(2*n+1)))^2)))} \\ Seiichi Manyama, Jan 02 2021

(Python)

# Using graphillion

from graphillion import GraphSet

def make_HOD(n):

    s = 1

    grids = []

    for i in range(2 * n + 1, 1, -2):

        for j in range(i - 2):

            a, b, c = s + j, s + j + 1, s + i + j

            grids.extend([(a, b), (b, c)])

        grids.append((s + i - 2, s + i - 1))

        s += i

    return grids

def A340185(n):

    if n == 0: return 1

    universe = make_HOD(n)

    GraphSet.set_universe(universe)

    spanning_trees = GraphSet.trees(is_spanning=True)

    return spanning_trees.len()

print([A340185(n) for n in range(7)])

CROSSREFS

Cf. A004003, A007725, A007726, A065072, A127605, A340052, A340176 (halved Aztec diamond HMD_n).

Sequence in context: A200797 A241116 A126678 * A208197 A232407 A208404

Adjacent sequences:  A340182 A340183 A340184 * A340186 A340187 A340188

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Dec 31 2020

STATUS

approved

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Last modified January 17 22:52 EST 2021. Contains 340247 sequences. (Running on oeis4.)