A340139 -id:A340139 - cp's OEIS frontend

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

1, 12, 17745, 2958176256, 54090331699622625, 107181043200192494332800000, 22868509031094388112997259982567521313, 523389340935243821042846225254323436248483571433472

Offset: 1

Seiichi Manyama, Dec 30 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..30
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, A340139, A340167.

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) ~ Gamma(1/4) * exp(8*G*n^2/Pi) / (Pi^(3/4) * sqrt(n) * 2^(6*n - 2)), where G is Catalan's constant A006752. - Vaclav Kotesovec, Jan 05 2021

A229728 Decimal expansion of the square of the constant A130834.

Original entry on oeis.org

3, 2, 0, 9, 9, 1, 2, 3, 0, 0, 7, 2, 8, 1, 5, 7, 6, 7, 8, 6, 2, 9, 7, 4, 9, 4, 8, 1, 7, 7, 9, 9, 0, 5, 1, 5, 8, 7, 4, 8, 5, 9, 2, 1, 2, 4, 2, 5, 1, 8, 3, 4, 4, 9, 4, 8, 7, 4, 5, 8, 6, 0, 0, 5, 8, 4, 6, 1, 0, 2, 4, 6, 4, 1, 6, 2, 4, 2, 4, 0, 2, 0, 4, 0, 6, 6, 7, 6, 7, 1, 2, 1, 5, 1, 4, 1, 0, 8, 8, 7, 0, 9, 4, 2, 8, 4, 6, 6, 9, 1, 5, 8, 3, 8, 7, 5, 2, 2, 6, 9

Offset: 1

N. J. A. Sloane, Oct 01 2013

			3.209912300728157678629749481779905158748592124251834494874586...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 232.

Cf. A000796, A006752, A097469, A130834.

Cf. A007341, A212800, A334088, A335586, A340139, A340176, A340557, A340562, A341479.

RealDigits[Exp[4*Catalan/Pi], 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)

exp(4*Catalan/Pi) \\ Charles R Greathouse IV, Jul 15 2014

From Amiram Eldar, Jun 12 2023: (Start)
Equals exp(4*G/Pi) = exp(4*A006752/A000796).
Equals A097469^4. (End)

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

Original entry on oeis.org

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

A340165 a(n) = 4^((n-2)(n-1)) Product_{1<=i

Original entry on oeis.org

1, 1, 19, 7056, 51251277, 7280323311888, 20225477546584790663, 1098876823994281426921193472, 1167619533875635661974056722756222809, 24263631353490502503207804571072304043237024000

Offset: 1

Seiichi Manyama, Dec 30 2020

nonn

Cf. A340139, A340167.

Table[4^((n-2)*(n-1)) * Product[Product[1 + Sin[i*Pi/(2*n)]^2 * Sin[j*Pi/(2*n)]^2, {i, 1, j-1}], {j, 2, n-1}], {n, 1, 12}] // Round (* Vaclav Kotesovec, Dec 31 2020 *)

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

a(n) = 4^((n-2)*(n-1)) * Product_{1<=i

a(n) ~ 2^(2*n^2 - 3*n + 35/8) * (1 - sqrt(2*sqrt(2)-2))^n * exp(2*A340350*n^2/Pi^2). - Vaclav Kotesovec, Jan 05 2021

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A340166 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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A229728 Decimal expansion of the square of the constant A130834.

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

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A340165 a(n) = 4^((n-2)*(n-1)) * Product_{1<=i

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

A340165 a(n) = 4^((n-2)(n-1)) Product_{1<=i