A370261 a(n) = sqrt(A370259(2*n)/(n+1)) for n >= 1.

1, 5, 65, 1449, 46561, 1968525, 103565057, 6531391313, 480749649601, 40482981221781, 3840053099665729, 405275779792031225, 47113209228513626017, 5982545638922153790749, 823992221632687352744961, 122360935410018418223907489, 19489013519781051891806113153

Offset: 1

Author

Peter Bala, Mar 11 2024

Keywords

• nonn
• easy

Comments

The sequence is conjectured to be integral.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A008310, A342205, A370259, A370260.

Programs

Maple

A370259 := n -> simplify( (ChebyshevT(n, n+1) - 1)/n^3 ):
seq(sqrt(A370259(2*n)/(n+1)), n = 1..20);

Mathematica

Table[Sqrt[(ChebyshevT[2*n, 2*n + 1] - 1)/(2*n)^3/(n + 1)], {n, 20}] (* Paolo Xausa, Jul 24 2024 *)

Python

from math import isqrt
from sympy import chebyshevt
def A370261(n): return isqrt((chebyshevt((m:=n<<1),m+1)-1)//((n+1)*m**3)) # Chai Wah Wu, Mar 13 2024

Formula

a(n) = sqrt( (T(2*n, 2*n+1) - 1)/((n+1)*(2*n)^3) ), where T(n, x) is the n-th Chebyshev polynomial of the first kind.