A188890 a(n) = 2^(n^2*2)*A004003(n).

1, 8, 9216, 1763704832, 55786539933761536, 291139753716719964894789632, 250571020614762113258116708732764160000, 35558299325468842404742419202148426327085023232000000, 831952537599396321581603077837672266625608464198409017861335832068096

Offset: 0

Author

N. J. A. Sloane, Apr 12 2011

Keywords

• nonn

Comments

The formula on line 3 of page 320 of the Levin et al. book has a typo: 2^(nm) should be 2^(mn/2). The main diagonal of the erroneous formula gives the present sequence.

References

• Levin, David A.; Peres, Yuval; and Wilmer, Elizabeth L.; Markov Chains and Mixing Times, American Mathematical Society, Providence, RI, 2009.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..30
• Wikipedia, Chebyshev polynomials

Crossrefs

Cf. A004003.

Programs

Maple

f:=(m,n)->
2^(m*n)*mul( mul(
(cos(Pi*i/(n+1))^2+cos(Pi*j/(m+1))^2), j=1..m/2), i=1..n/2);
g:=(m,n)->round(evalf(f(m,n),300));
t1:=[seq(g(2*n,2*n),n=0..10)];

Mathematica

Table[Sqrt[Resultant[ChebyshevU[2*n, x], ChebyshevU[2*n, I*x], x]], {n, 0, 10}] (* Vaclav Kotesovec, Apr 14 2020 *)

PARI

{a(n) = sqrtint(polresultant(polchebyshev(2*n, 2, x), polchebyshev(2*n, 2, I*x)))} \\ Seiichi Manyama, Apr 14 2020

Python

from math import isqrt
from sympy import resultant, chebyshevu, I
from sympy.abc import x
def A188890(n): return isqrt(resultant(chebyshevu(n<<1,x),chebyshevu(n<<1,I*x))) if n else 1 # Chai Wah Wu, Nov 08 2023

Formula

a(n)^2 = Resultant(U(2*n,x), U(2*n,i*x)), where U(n,x) is a Chebyshev polynomial of the second kind and i = sqrt(-1). - Seiichi Manyama, Apr 14 2020

a(n) ~ 2^(2*n^2 + 1) * (sqrt(2)-1)^(2*n+1) * exp(G*(2*n+1)^2/Pi), where G is Catalan's constant A006752. - Vaclav Kotesovec, Apr 14 2020, updated Jan 03 2021