A334088 -id:A334088 - cp's OEIS frontend

A334089 a(n) = sqrt(A334088(n)/2^(n-1)).

1, 2, 13, 272, 18281, 3944920, 2732887529, 6077512159232, 43384923739812577, 994156445200670735008, 73125714588602035608260981, 17265651822746410593596262486016, 13085551252412040683513520733767180041, 31834381760532514451976501491991780699626368

Offset: 1

Seiichi Manyama, Apr 14 2020

nonn

Wikipedia, Chebyshev polynomials
Wikipedia, Resultant

Cf. A065072, A334088, A340295.

Table[Resultant[ChebyshevT[2*n, x/2], ChebyshevT[2*n, I*x/2], x]^(1/4) / 2^((n-1)/2), {n, 1, 15}] (* Vaclav Kotesovec, Apr 14 2020 *)

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

a(n) ~ exp(2*G*n^2/Pi) / 2^(3*n/2 - 5/8), where G is Catalan's constant A006752. - Vaclav Kotesovec, Apr 14 2020

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

A334124 a(n) = 2^n * sqrt(Resultant(U_{2n}(x/2), T_{2n}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

Original entry on oeis.org

1, 3, 71, 17753, 46069729, 1234496016491, 341133743251787719, 971684488369988888850993, 28523907708086181923163934073729, 8628515016553040037389969912341438652243, 26895841132028233579514694272575933932911355677831

Offset: 0

Seiichi Manyama, Apr 15 2020

nonn

Wikipedia, Chebyshev polynomials
Wikipedia, Resultant

Main diagonal of A103997.

Cf. A004003, A334088.

Table[2^n * Sqrt[Resultant[ChebyshevU[2*n, x/2], ChebyshevT[2*n, I*x/2], x]], {n, 0, 12}] (* Vaclav Kotesovec, Apr 16 2020 *)

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

from math import isqrt
from sympy import resultant, chebyshevt, chebyshevu, I
from sympy.abc import x
def A334124(n): return isqrt(resultant(chebyshevu(n<<1,x/2),chebyshevt(n<<1,I*x/2))*(1<<(n<<1))) if n else 1 # Chai Wah Wu, Nov 07 2023

a(n) = A103997(n,n).

a(n) ~ 2^(1/4) * exp(2*G*n*(2*n+1)/Pi) / (1 + sqrt(2))^n, where G is Catalan's constant A006752. - Vaclav Kotesovec, Apr 16 2020, updated Jan 03 2021

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A334089 a(n) = sqrt(A334088(n)/2^(n-1)).

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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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A334124 a(n) = 2^n * sqrt(Resultant(U_{2*n}(x/2), T_{2*n}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

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Python

Formula

A334124 a(n) = 2^n * sqrt(Resultant(U_{2n}(x/2), T_{2n}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).