A173127 -id:A173127 - cp's OEIS frontend

A173129 a(n) = cosh(2 * n * arccosh(n)).

1, 1, 97, 19601, 7380481, 4517251249, 4097989415521, 5170128475599457, 8661355881006882817, 18605234632923999244961, 49862414878754347585980001, 163104845048002042971670685041, 639582975902942936737758325440001

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..193
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

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

Cf. A349070, A349071, A349073.

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

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 *)

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

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

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

A173128 a(n) = cosh(2narcsinh(n)).

Original entry on oeis.org

1, 3, 161, 27379, 9478657, 5517751251, 4841332221601, 5964153172084899, 9814664424981012481, 20791777842234580902499, 55106605639755476546020001, 178627672869645203363556318483, 695165908550906808156689590141441

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Robert Israel, Table of n, a(n) for n = 0..192
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173129, A173174.

seq(expand( (1/2)*((n + sqrt(n^2 + 1))^(2*n) + (n - sqrt(n^2 + 1))^(2*n))), n=0..30); # Robert Israel, Apr 05 2016

Round[Table[Cosh[2 n ArcSinh[n]], {n, 0, 20}]] (* Artur Jasinski *)
Round[Table[1/2 (x - Sqrt[1 + x^2])^(2 x) + 1/2 (x + Sqrt[1 + x^2])^(2 x), {x, 0, 20}]] (* Artur Jasinski, Feb 14 2010 *)

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

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

A173130 a(n) = Cosh[(2 n - 1) ArcCosh[n]].

Original entry on oeis.org

0, 1, 26, 3363, 937444, 456335045, 343904160606, 371198523608647, 543466014742175624, 1036834190110356583689, 2499384905955651114739810, 7429238104512325157021090411, 26695718139185294187938997247212

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128.

Table[Round[Cosh[(2 n - 1) ArcCosh[n]]], {n, 0, 20}] (* Artur Jasinski *)

a(n) ~ 2^(2*n-2) * n^(2*n-1). - Vaclav Kotesovec, Apr 05 2016

A173131 a(n) = (Cosh[(2n-1)ArcSinh[n]])^2.

Original entry on oeis.org

1, 2, 1445, 19740250, 1361599599377, 298514762397852026, 160545187370375075046277, 179656719395983409634002348450, 373368546362937441101158606899394625

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130.

Table[Round[Cosh[(2 n - 1) ArcSinh[n]]^2], {n, 0, 10}] (* Artur Jasinski *)

a(n) ~ 2^(4*n-4) * n^(4*n-2). - Vaclav Kotesovec, Apr 05 2016

A173148 a(n) = cos(2narccos(sqrt(n))).

Original entry on oeis.org

1, 1, 17, 485, 18817, 930249, 55989361, 3974443213, 325142092801, 30122754096401, 3117419602578001, 356452534779818421, 44627167107085622401, 6071840759403431812825, 892064955046043465408177, 140751338790698080509966749, 23737154316161495960243527681

Offset: 0

Artur Jasinski, Feb 11 2010

nonn

The Chebyshev polynomial T_n is defined by cos(nx) = T_n(cos(x)). So T_2n(cos(x)) = cos(2nx) = cos^2(nx) - 1 = (T_n(x))^2 - 1 consists of only even powers of x. As a result, a(n) = T_2n(sqrt(n)) is an integer. - Michael B. Porter, Jan 01 2019

Seiichi Manyama, Table of n, a(n) for n = 0..321
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Cf. A053120 (Chebyshev polynomial), A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A322790.

a:=List([0..20],n->Sum([0..n],k->Binomial(2*n,2*k)*(n-1)^(n-k)*n^k));; Print(a); # Muniru A Asiru, Jan 03 2019

[&+[Binomial(2*n,2*k)*(n-1)^(n-k)*n^k: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Jan 03 2019

Table[Round[Cos[2 n ArcCos[Sqrt[n]]]], {n, 0, 30}] (* Artur Jasinski, Feb 11 2010 *)

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

{a(n) = round(cosh(2*n*acosh(sqrt(n))))} \\ Seiichi Manyama, Dec 27 2018

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

a(n) ~ exp(-1/2) * 2^(2*n-1) * n^n. - Vaclav Kotesovec, Apr 05 2016
a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n-1)^(n-k)*n^k. - Seiichi Manyama, Dec 27 2018
a(n) = cosh(2*n*arccosh(sqrt(n))). - Seiichi Manyama, Dec 27 2018
a(n) = T_{2*n}(sqrt(n)) = T_{n}(2*n-1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018
a(n) = A322790(n-1, n) for n > 0. - Seiichi Manyama, Dec 29 2018

Minor edits by Vaclav Kotesovec, Apr 05 2016

A173133 a(n) = Sinh[(2n-1) ArcSinh[n]].

Original entry on oeis.org

0, 1, 38, 4443, 1166876, 546365045, 400680904674, 423859315570607, 611038907405197432, 1151555487914640463209, 2748476184146759127540190, 8102732939160371170806346243, 28915133156938367486730067779348

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131.

Table[Round[Sinh[(2 n - 1) ArcSinh[n]]], {n, 0, 20}] (* Artur Jasinski *)
Round[Table[1/2 (n - Sqrt[1 + n^2])^(2 n - 1) + 1/2 (n + Sqrt[1 + n^2])^(2 n - 1), {n, 0, 10}]] (* Artur Jasinski, Feb 14 2010 *)

a(n) = 1/2 (n - sqrt(1 + n^2))^(2 n - 1) + 1/2 (n + sqrt(1 + n^2))^(2 n - 1). - Artur Jasinski, Feb 14 2010

Minor edits by Vaclav Kotesovec, Apr 05 2016

A173134 a(n) = Sinh[(2n-1)ArcCosh[n]]^2.

Original entry on oeis.org

-1, 0, 675, 11309768, 878801253135, 208241673295152024, 118270071682117442287235, 137788343929239264227213170608, 295355309179742652677310128859789375

Offset: 0

Artur Jasinski, Feb 10 2010

sign

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133.

Table[Round[Sinh[(2 n - 1) ArcCosh[n]]^2], {n, 0, 20}]

a(n) ~ 2^(4*n-4) * n^(4*n-2). - Vaclav Kotesovec, Apr 05 2016

A173170 a(n) = sin^2((2n-1)arcsin(sqrt n)) = 1 - sin^2( (2n-1)arccos(sqrt n)).

Original entry on oeis.org

0, 1, 50, 23763, 25421764, 48225038405, 142786923879606, 608447515452613207, 3527836867501829594888, 26710782540478226038759689, 255922222218837615280903143610, 3026917140685147530327256796600411

Offset: 0

Artur Jasinski, Feb 11 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..170

Cf. A132592, A146311, A146312, A146313, A173115, A173116 A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151.

Table[Round[Sin[(2 n - 1) ArcSin[Sqrt[n]]]^2], {n, 0, 20}] (* Artur Jasinski, Feb 11 2010 *)

a(n) ~ exp(-1) * 2^(4*n-4) * n^(2*n-1). - Vaclav Kotesovec, Apr 05 2016

Minor edits by Vaclav Kotesovec, Apr 05 2016

A173174 a(n) = cosh(2narcsinh(sqrt(n))).

Original entry on oeis.org

1, 3, 49, 1351, 51841, 2550251, 153090001, 10850138895, 886731088897, 82094249361619, 8491781781142001, 970614726270742103, 121485428812828080001, 16525390478051500325307, 2427469037137019032095121, 382956978214541873571486751, 64576903826545426454350012417, 11591229031806966336496244914595

Offset: 0

Artur Jasinski, Feb 11 2010

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..321
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Cf. A132592, A146311 - A146313, A173115, A173116 A173121, A173127 - A173131, A173133, A173134, A173148, A173151, A173170, A173171.
Cf. A322746.
Main diagonal of A322790.

[&+[Binomial(2*n, 2*k)*(n+1)^(n-k)*n^k: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Dec 29 2018

A173174 := proc(n) cosh(2*n*arcsinh(sqrt(n))) ; expand(%) ; simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011

Table[Round[N[Cosh[(2 n) ArcSinh[Sqrt[n]]], 100]], {n, 0, 30}] (* Artur Jasinski *)
Join[{1}, a[n_]:=Sum[Binomial[2 n, 2 k] (n + 1)^(n - k) n^k, {k, 0, n}]; Array[a, 25]] (* Vincenzo Librandi, Dec 29 2018 *)

{a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n+1)^(n-k)*n^k)} \\ Seiichi Manyama, Dec 26 2018

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

a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n+1)^(n-k)*n^k. - Seiichi Manyama, Dec 26 2018

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

More terms from Seiichi Manyama, Dec 26 2018

A173171 a(n) = - sin^2((2n-1)arccos(sqrt n)) = sin^2((2n-1)arcsin(sqrt n)) - 1.

Original entry on oeis.org

-1, 0, 49, 23762, 25421763, 48225038404, 142786923879605, 608447515452613206, 3527836867501829594887, 26710782540478226038759688, 255922222218837615280903143609, 3026917140685147530327256796600410

Offset: 0

Artur Jasinski, Feb 11 2010

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..170

Cf. A132592, A146311, A146312, A146313, A173115, A173116 A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151, A173170.

Table[Round[-N[ Sin[(2 n - 1) ArcCos[Sqrt[n]]]^2, 100]], {n, 0, 20}] (* Artur Jasinski, Feb 11 2010; Typo fixed by Vincenzo Librandi, Jun 29 2014 *)

cp's OEIS Frontend

A173129 a(n) = cosh(2 * n * arccosh(n)).

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A173128 a(n) = cosh(2*n*arcsinh(n)).

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A173130 a(n) = Cosh[(2 n - 1) ArcCosh[n]].

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A173131 a(n) = (Cosh[(2n-1)ArcSinh[n]])^2.

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A173148 a(n) = cos(2*n*arccos(sqrt(n))).

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A173133 a(n) = Sinh[(2n-1) ArcSinh[n]].

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A173134 a(n) = Sinh[(2n-1)ArcCosh[n]]^2.

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A173170 a(n) = sin^2((2n-1)*arcsin(sqrt n)) = 1 - sin^2( (2n-1)*arccos(sqrt n)).

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A173128 a(n) = cosh(2narcsinh(n)).

A173148 a(n) = cos(2narccos(sqrt(n))).

A173170 a(n) = sin^2((2n-1)arcsin(sqrt n)) = 1 - sin^2( (2n-1)arccos(sqrt n)).

A173174 a(n) = cosh(2narcsinh(sqrt(n))).

A173171 a(n) = - sin^2((2n-1)arccos(sqrt n)) = sin^2((2n-1)arcsin(sqrt n)) - 1.