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A173129 a(n) = cosh(2 * n * arccosh(n)). 13
1, 1, 97, 19601, 7380481, 4517251249, 4097989415521, 5170128475599457, 8661355881006882817, 18605234632923999244961, 49862414878754347585980001, 163104845048002042971670685041, 639582975902942936737758325440001 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..193

Wikipedia, Chebyshev polynomials.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n) = (1/2)*((n+sqrt(n^2-1))^(2*n) + (n-sqrt(n^2-1))^(2*n)). - Artur Jasinski, Feb 14 2010, corrected by Vaclav Kotesovec, Apr 05 2016

a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n^2-1)^(n-k)*n^(2*k). - Seiichi Manyama, Dec 27 2018

a(n) = T_{2n}(n) where T_{2n} is a Chebyshev polynomial of the first kind. - Robert Israel, Dec 27 2018

a(n) = T_{n}(2*n^2-1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018

MAPLE

seq(orthopoly[T](2*n, n), n=0..50); # Robert Israel, Dec 27 2018

MATHEMATICA

Table[Round[Cosh[2 n ArcCosh[n]]], {n, 0, 20}] (* Artur Jasinski, Feb 10 2010 *)

Round[Table[1/2 (x - Sqrt[ -1 + x^2])^(2 x) + 1/2 (x + Sqrt[ -1 + x^2])^(2 x), {x, 0, 10}]] (* Artur Jasinski, Feb 14 2010 *)

PROG

(PARI) a(n) = round(cosh(2*n*acosh(n))); \\ Michel Marcus, Apr 05 2016

(PARI) {a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n^2-1)^(n-k)*n^(2*k))} \\ Seiichi Manyama, Dec 27 2018

(PARI) {a(n) = polchebyshev(2*n, 1, n)} \\ Seiichi Manyama, Dec 28 2018

(PARI) {a(n) = polchebyshev(n, 1, 2*n^2-1)} \\ Seiichi Manyama, Dec 29 2018

CROSSREFS

Cf. A001079, A037270, A053120 (Chebyshev polynomial), A058331, A115066, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173148.

Sequence in context: A219062 A218318 A233426 * A321041 A173354 A176135

Adjacent sequences:  A173126 A173127 A173128 * A173130 A173131 A173132

KEYWORD

nonn

AUTHOR

Artur Jasinski, Feb 10 2010

STATUS

approved

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Last modified April 19 14:14 EDT 2019. Contains 322280 sequences. (Running on oeis4.)