A370259 a(n) = (T(n,n+1) - 1)/n^3 for n >= 1, where T(n,x) is the n-th Chebyshev polynomial of the first kind.

1, 2, 9, 75, 961, 16900, 380689, 10498005, 343323841, 13007560326, 560789801881, 27125634729375, 1455389462287489, 85805768251305992, 5515372218107327521, 383931652351786775721, 28778117694539885440129, 2311202255914842794592010, 198009919900727928789497641, 18027589454633803742596931571

Offset

1,2

Comments

It appears that a(2*n+1) is always a square, while a(2*n) = (n + 1) * a square. See A370260 and A370261.

Links

Paolo Xausa, Table of n, a(n) for n = 1..350

Formula

a(n) = Sum_{k = 1..n} (2^k)*n^(k-2)*binomial(n+k, 2*k)/(n + k) (shows that a(n) is an integer).

a(n) = (cos(n*arccos(n+1)) - 1)/n^3.

a(n) = (A342205(n) - 1)/n^3.

a(n) = ( (n + 1 + sqrt(n*(n+2)))^n + (n + 1 - sqrt(n*(n+2)))^n - 2 )/(2*n^3).

Maple

seq( simplify( (ChebyshevT(n, n+1) - 1)/n^3 ), n = 1..20);

Mathematica

Array[(ChebyshevT[#, #+1]-1)/#^3 &, 20] (* Paolo Xausa, Mar 14 2024 *)

Prog

(Python)
from sympy import chebyshevt
def A370259(n): return (chebyshevt(n, n+1)-1)//n**3 # Chai Wah Wu, Mar 13 2024

Crossrefs

Cf. A008310, A342205, A370260, A370261.

Sequence in context: A197082 A243054 A080638 * A335378 A232471 A357539

Adjacent sequences: A370256 A370257 A370258 * A370260 A370261 A370262

Keyword

nonn,easy

Author

Peter Bala, Mar 11 2024

Status

approved