A340166 -id:A340166 - cp's OEIS frontend

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

1, 1, 4, 208, 121856, 772189440, 51989627289600, 36837279603595907072, 273129993621426778551615488, 21114078836429317912110529666154496, 16975032309392309949804839529585109326888960

Offset: 0

Seiichi Manyama, Dec 31 2020

nonn

                                  *---*
                                  |   |
              *---*           *---*---*---*
              |   |           |   |   |   |
  *---*   *---*---*---*   *---*---*---*---*---*
  HMD_1       HMD_2               HMD_3
-------------------------------------------------
              *---*
              |   |
          *---*---*---*
          |   |   |   |
      *---*---*---*---*---*
      |   |   |   |   |   |
  *---*---*---*---*---*---*---*
              HMD_4

			a(2) = 4;
      *   *           *---*           *---*           *---*
      |   |               |           |               |   |
  *---*---*---*   *---*---*---*   *---*---*---*   *---*   *---*

Seiichi Manyama, Table of n, a(n) for n = 0..45
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 3.6.
Wikipedia, Chebyshev polynomials
Wikipedia, Resultant

Cf. A007341, A007725, A007726, A334088, A334089, A340139, A340166, A340185 (halved Aztec diamond HOD_n).

default(realprecision, 120);
{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)))))}

{a007341(n) = polresultant(polchebyshev(n-1, 2, x/2), polchebyshev(n-1, 2, (4-x)/2))};
{a334088(n) = sqrtint(polresultant(polchebyshev(2*n, 1, x/2), polchebyshev(2*n, 1, I*x/2)))};
{a(n) = if(n==0, 1, sqrtint(a007341(n)*a334088(n)/n))}

default(realprecision, 120);
{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

# Using graphillion
from graphillion import GraphSet
def make_HMD(n):
    s = 1
    grids = []
    for i in range(2 * n, 0, -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 A340176(n):
    if n == 0: return 1
    universe = make_HMD(n)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
print([A340176(n) for n in range(7)])

a(n) = Product_{1<=j

a(n) = 2^(n-1) * A007726(n) * A334089(n) = sqrt(A007341(n) * A334088(n) / n) for n > 0.

a(n) = 4^(n-1) * A340139(n) = 4^((n-1)^2) * Product_{1<=j 0. - Seiichi Manyama, Jan 02 2021

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

A340291 a(n) = 4^(2n^2) Product_{1<=j,k<=n} (1 - cos(jPi/(2n+1))^2 * cos(kPi/(2n+1))^2).

Original entry on oeis.org

1, 15, 32625, 8238791743, 230629380093001665, 703130165949449759361247759, 231459008314298532714943209968328640625, 8186710889725936196671113787217620194601044287109375

Offset: 0

Seiichi Manyama, Jan 03 2021

nonn

Cf. A093967, A340166, A340185, A340292, A340295.

Table[2^(4*n^2) * Product[Product[1 - Cos[j*Pi/(2*n+1)]^2 * Cos[k*Pi/(2*n+1)]^2, {j, 1, n}], {k, 1, n}], {n, 0, 10}] // Round (* Vaclav Kotesovec, Jan 03 2021 *)

default(realprecision, 120);
{a(n) = round(4^(2*n^2)*prod(j=1, n, prod(k=1, n, 1-(cos(j*Pi/(2*n+1))*cos(k*Pi/(2*n+1)))^2)))}

a(n) = A093967(2*n+1) * A340185(n)^2.

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

A340292 a(n) = 4^(2n^2) Product_{1<=j,k<=n} (1 - sin(jPi/(2n+1))^2 * sin(kPi/(2n+1))^2).

Original entry on oeis.org

1, 7, 4961, 371647151, 2952717950351617, 2489597262406609716450871, 222812636926792555435326125877303201, 2116840405025957772469476908228785308996001314527, 2134958300495920487325052422663717579194357002081033470045923329

Offset: 0

Seiichi Manyama, Jan 03 2021

nonn

Cf. A002315, A340166, A340291, A340293, A340295.

Table[2^(4*n^2) * Product[Product[1 - Sin[j*Pi/(2*n + 1)]^2 * Sin[k*Pi/(2*n + 1)]^2, {k, 1, n}], {j, 1, n}], {n, 0, 10}] // Round (* Vaclav Kotesovec, Jan 04 2021 *)

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

a(n) = A002315(n) * A340293(n)^2.

a(n) ~ exp(2*G*(2*n+1)^2/Pi) / 2^(4*n + 3/4), where G is Catalan's constant A006752. - Vaclav Kotesovec, Jan 04 2021

A340295 a(n) = 4^(2n^2) Product_{1<=j,k<=n} (1 - sin(jPi/(2n+1))^2 * cos(kPi/(2n+1))^2).

Original entry on oeis.org

1, 13, 18281, 2732887529, 43384923739812577, 73125714588602035608260981, 13085551252412040683513520733767180041, 248596840858215958581954513797323868183183928594833

Offset: 0

Seiichi Manyama, Jan 03 2021

nonn

a(n)/A001570(n+1) is an integer.

Wikipedia, Chebyshev polynomials
Wikipedia, Resultant

Cf. A001570, A334089, A340166, A340291, A340292.

Table[Resultant[ChebyshevT[4*n+2, x/2], ChebyshevT[4*n+2, I*x/2], x]^(1/4) / 2^n, {n, 0, 10}] (* Vaclav Kotesovec, Jan 04 2021 *)

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

{a(n) = sqrtint(sqrtint(polresultant(polchebyshev(4*n+2, 1, x/2), polchebyshev(4*n+2, 1, I*x/2))))/2^n}

a(n) = A334089(2*n+1).

a(n) ~ exp(2*G*(2*n+1)^2/Pi) / 2^(3*n + 7/8), where G is Catalan's constant A006752. - Vaclav Kotesovec, Jan 04 2021

A340427 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = 4^(2(n-1)(k-1)) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(aPi/(2n))^2 * sin(bPi/(2k))^2).

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 140, 140, 1, 1, 1632, 17745, 1632, 1, 1, 19024, 2227120, 2227120, 19024, 1, 1, 221760, 279215849, 2958176256, 279215849, 221760, 1, 1, 2585024, 35001302700, 3909096873216, 3909096873216, 35001302700, 2585024, 1

Offset: 1

Seiichi Manyama, Jan 07 2021

			Square array begins:
  1,     1,         1,             1,                 1, ...
  1,    12,       140,          1632,             19024, ...
  1,   140,     17745,       2227120,         279215849, ...
  1,  1632,   2227120,    2958176256,     3909096873216, ...
  1, 19024, 279215849, 3909096873216, 54090331699622625, ...

Main diagonal gives A340166.

Cf. A340425, A340428, A340430, A340432.

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

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

A340167 a(n) = 4^(2(n-1)^2) Product_{1<=i,j<=n-1} (1 + sin(iPi/(2n))^2 * sin(jPi/(2n))^2).

Original entry on oeis.org

1, 20, 153425, 450075709440, 504979178328238519521, 216703205118496785026106198144000, 35568160616301682717925992221900586646216066081, 2232861039051291914755952483706805051795013026559178904468193280

Offset: 1

Seiichi Manyama, Dec 30 2020

nonn

Cf. A340165, A340166.

Table[4^(2*(n-1)^2) * Product[Product[1 + Sin[i*Pi/(2*n)]^2 * Sin[j*Pi/(2*n)]^2, {i, 1, n-1}], {j, 1, n-1}], {n, 1, 10}] // Round (* Vaclav Kotesovec, Dec 31 2020 *)

default(realprecision, 120);
{a(n) = round(4^(2*(n-1)^2)*prod(i=1, n-1, prod(j=1, n-1, 1+(sin(i*Pi/(2*n))*sin(j*Pi/(2*n)))^2)))}

a(n) = 4^(2*(n-1)^2) * Product_{1<=i,j<=n-1} (1 + cos(i*Pi/(2*n))^2 * cos(j*Pi/(2*n))^2).
a(n) = 4^(2*(n-1)^2) * Product_{1<=i,j<=n-1} (1 + sin(i*Pi/(2*n))^2 * cos(j*Pi/(2*n))^2).
a(n) ~ 2^(4*n^2 - 6*n + 17/4) * (sqrt(2) - 1)^(2*n) * exp(4*A340350*n^2/Pi^2). - Vaclav Kotesovec, Jan 05 2021

A340352 Number of spanning trees of odd Aztec diamond OD_n.

Original entry on oeis.org

1, 192, 4542720, 12116689944576, 3544863978266468352000, 112387469554685044937510092800000, 383669915612621265759587438135691539652804608, 140496256399491641572818822014023027580848616806252629983232

Offset: 1

Seiichi Manyama, Jan 05 2021

nonn

R. P. Stanley conjectured that the even Aztec diamond has exactly four times as many spanning trees as the odd Aztec diamond. This conjecture was first proved by D. E. Knuth.
                                              *
                                              |
                      *                   *---*---*
                      |                   |   |   |
      *           *---*---*           *---*---*---*---*
      |           |   |   |           |   |   |   |   |
  *---*---*   *---*---*---*---*   *---*---*---*---*---*---*
      |           |   |   |           |   |   |   |   |
      *           *---*---*           *---*---*---*---*
                      |                   |   |   |
                      *                   *---*---*
                                              |
                                              *
     OD_1            OD_2                    OD_3

D. E. Knuth, Aztec Diamonds, Checkerboard Graphs, and Spanning Trees, arXiv:math/9501234 [math.CO], 1995; J. Alg. Combinatorics 6 (1997), 253-257.

Cf. A007725 (even Aztec diamond), A340166, A340185 (halved Aztec diamond HOD_n).

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

a(n) = 4^(2*(n-1)) * A340166(n) = 4^(2*(n-1)*n) * Product_{1<=j,k<=n-1} (1 - sin(j*Pi/(2*n))^2 * sin(k*Pi/(2*n))^2).

a(n) ~ Gamma(1/4) * exp(8*G*n^2/Pi) / (Pi^(3/4) * sqrt(n) * 2^(2*n + 2)), where G is Catalan's constant A006752. - Vaclav Kotesovec, Jan 06 2021

A071102 Determinant of KK* where K is Kasteleyn-Percus matrix for fool's diamond of order n.

Original entry on oeis.org

1, 2, 15, 384, 32625, 9085440, 8238791743, 24233379889152

Offset: 1

N. J. A. Sloane, May 28 2002

J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 27).

J. Propp, Updated article
J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics

Cf. A340166, A340291.

Conjecture from Seiichi Manyama, Jan 04 2021: (Start)
a(2*n+1) = A340291(n) = 4^(2*n^2) * Product_{1<=j,k<=n} (1 - cos(j*Pi/(2*n+1))^2 * cos(k*Pi/(2*n+1))^2).
a(2*n) = 2 * 4^(2*(n-1)) * A340166(n) = 2 * 4^(2*(n-1)*n) * Product_{1<=j,k<=n-1} (1 - cos(j*Pi/(2*n))^2 * cos(k*Pi/(2*n))^2). (End)

Showing 1-9 of 9 results.

cp's OEIS Frontend

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

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

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A340291 a(n) = 4^(2*n^2) * Product_{1<=j,k<=n} (1 - cos(j*Pi/(2*n+1))^2 * cos(k*Pi/(2*n+1))^2).

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A340292 a(n) = 4^(2*n^2) * Product_{1<=j,k<=n} (1 - sin(j*Pi/(2*n+1))^2 * sin(k*Pi/(2*n+1))^2).

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A340295 a(n) = 4^(2*n^2) * Product_{1<=j,k<=n} (1 - sin(j*Pi/(2*n+1))^2 * cos(k*Pi/(2*n+1))^2).

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A340427 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = 4^(2*(n-1)*(k-1)) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(a*Pi/(2*n))^2 * sin(b*Pi/(2*k))^2).

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A340167 a(n) = 4^(2*(n-1)^2) * Product_{1<=i,j<=n-1} (1 + sin(i*Pi/(2*n))^2 * sin(j*Pi/(2*n))^2).

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A340352 Number of spanning trees of odd Aztec diamond OD_n.

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A340291 a(n) = 4^(2n^2) Product_{1<=j,k<=n} (1 - cos(jPi/(2n+1))^2 * cos(kPi/(2n+1))^2).

A340292 a(n) = 4^(2n^2) Product_{1<=j,k<=n} (1 - sin(jPi/(2n+1))^2 * sin(kPi/(2n+1))^2).

A340295 a(n) = 4^(2n^2) Product_{1<=j,k<=n} (1 - sin(jPi/(2n+1))^2 * cos(kPi/(2n+1))^2).

A340427 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = 4^(2(n-1)(k-1)) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(aPi/(2n))^2 * sin(bPi/(2k))^2).

A340167 a(n) = 4^(2(n-1)^2) Product_{1<=i,j<=n-1} (1 + sin(iPi/(2n))^2 * sin(jPi/(2n))^2).