A173129
a(n) = cosh(2 * n * arccosh(n)).
Original entry on oeis.org
1, 1, 97, 19601, 7380481, 4517251249, 4097989415521, 5170128475599457, 8661355881006882817, 18605234632923999244961, 49862414878754347585980001, 163104845048002042971670685041, 639582975902942936737758325440001
Offset: 0
Cf.
A001079,
A037270,
A053120 (Chebyshev polynomial),
A058331,
A115066,
A132592,
A146311,
A146312,
A146313,
A173115,
A173116,
A173121,
A173127,
A173128,
A173148.
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seq(orthopoly[T](2*n,n), n=0..50); # Robert Israel, Dec 27 2018
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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 *)
Table[ChebyshevT[2*n, n], {n, 0, 15}] (* Vaclav Kotesovec, Nov 07 2021 *)
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{a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n^2-1)^(n-k)*n^(2*k))} \\ Seiichi Manyama, Dec 27 2018
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{a(n) = polchebyshev(2*n, 1, n)} \\ Seiichi Manyama, Dec 28 2018
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{a(n) = polchebyshev(n, 1, 2*n^2-1)} \\ Seiichi Manyama, Dec 29 2018
A115066
Chebyshev polynomial of the first kind T(n,x), evaluated at x=n.
Original entry on oeis.org
1, 1, 7, 99, 1921, 47525, 1431431, 50843527, 2081028097, 96450076809, 4993116004999, 285573847759211, 17882714781360001, 1216895030905226413, 89415396036432386311, 7055673735003659189775, 595077930963909484707841, 53421565080956452077519377
Offset: 0
a(3) = 99 because T[3, x] = 4x^3 - 3x and T[3, 3] = 4*3^3 - 3*3 = 99.
- G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1966, p. 35.
- M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 18.
- G. Szego, Orthogonal polynomials, Amer. Math. Soc., Providence, 1939, p. 29.
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with(orthopoly): seq(T(n,n),n=0..17);
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Table[ChebyshevT[n, n], {n, 0, 17}] (* Arkadiusz Wesolowski, Nov 17 2012 *)
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A115066(n)=cos(n*acos(n)) \\ M. F. Hasler, Apr 06 2012
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a(n) = polchebyshev(n, 1, n); \\ Seiichi Manyama, Dec 28 2018
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a(n) = if(n==0, 1, n*sum(k=0, n, (2*n-2)^k*binomial(n+k, 2*k)/(n+k))); \\ Seiichi Manyama, Mar 05 2021
A349073
a(n) = U(2*n, n), where U(n, x) is the Chebyshev polynomial of the second kind.
Original entry on oeis.org
1, 3, 209, 40391, 15003009, 9127651499, 8254109243953, 10393834843080975, 17391182043967249409, 37326390852372133364819, 99976027392046047055178001, 326887883645157139828711692503, 1281398359905415379814555044995201, 5932135472283024519893762690145006075
Offset: 0
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Table[ChebyshevU[2*n, n], {n, 0, 15}]
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a(n) = polchebyshev(2*n, 2, n); \\ Michel Marcus, Nov 07 2021
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from sympy import chebyshevu
def A349073(n): return chebyshevu(n<<1,n) # Chai Wah Wu, Nov 08 2023
A349072
a(n) = T(n, 3*n), where T(n, x) is the Chebyshev polynomial of the first kind.
Original entry on oeis.org
1, 3, 71, 2889, 164737, 12082575, 1083358151, 114812765781, 14040770918401, 1946133989077851, 301491888156044999, 51624542295308885793, 9681761035138427706241, 1973656779656041723763559, 434528364117341972641648967, 102755067271708508826774929325
Offset: 0
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Table[ChebyshevT[n, 3*n], {n, 0, 20}]
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a(n) = polchebyshev(n, 1, 3*n); \\ Michel Marcus, Nov 07 2021
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from sympy import chebyshevt
def A349072(n): return chebyshevt(n,3*n) # Chai Wah Wu, Nov 08 2023
A349076
a(n) = U(n, 3*n), where U(n, x) is the Chebyshev polynomial of the second kind.
Original entry on oeis.org
1, 6, 143, 5796, 330049, 24192090, 2168392031, 229755926568, 28093745899009, 3893604149949966, 603151411514453999, 103272803655639197580, 19367259480582106560193, 3947962681769909551857186, 869179946261864224288867775, 205535515565731164929726435280
Offset: 0
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Table[ChebyshevU[n, 3*n], {n, 0, 20}]
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a(n) = polchebyshev(n, 2, 3*n); \\ Michel Marcus, Nov 07 2021
Showing 1-5 of 5 results.
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