A115066 -id:A115066 - 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

A323118 a(n) = U_{n}(n) where U_{n}(x) is a Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 2, 15, 204, 3905, 96030, 2883167, 102213944, 4178507265, 193501094490, 10011386405999, 572335117886532, 35827847605137601, 2437406399741075126, 179059769134174484415, 14127079203550978667760, 1191321539697176278429697, 106935795565608726499866930

Offset: 0

Seiichi Manyama, Jan 05 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..351
Wikipedia, Chebyshev polynomials.

Main diagonal of A323182.

Cf. A115066, A318192, A323117, A349073, A349074, A349075, A349076.

Table[ChebyshevU[n, n], {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)

PARI
```
a(n) = polchebyshev(n, 2, n);
```

a(n) = sum(k=0, n\2, (n^2-1)^k*n^(n-2*k)*binomial(n+1, 2*k+1));

a(n) = sum(k=0, n, (2*n-2)^k*binomial(n+1+k, 2*k+1)); \\ Seiichi Manyama, Mar 03 2021

a(n) = Sum_{k=0..floor(n/2)} (n^2-1)^k*n^(n-2*k) * binomial(n+1,2*k+1).
a(n) ~ 2^n * n^n. - Vaclav Kotesovec, Jan 05 2019
a(n) = Sum_{k=0..n} (2*n-2)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (2*n-2)^k * binomial(n+1+k,2*k+1). - Seiichi Manyama, Mar 03 2021

A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

Original entry on oeis.org

1, 1, 0, 1, 1, -1, 1, 2, 1, 0, 1, 3, 7, 1, 1, 1, 4, 17, 26, 1, 0, 1, 5, 31, 99, 97, 1, -1, 1, 6, 49, 244, 577, 362, 1, 0, 1, 7, 71, 485, 1921, 3363, 1351, 1, 1, 1, 8, 97, 846, 4801, 15124, 19601, 5042, 1, 0, 1, 9, 127, 1351, 10081, 47525, 119071, 114243, 18817, 1, -1

Offset: 0

Seiichi Manyama, Dec 28 2018

			Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 1,    2,     3,      4,      5,       6, ...
  -1, 1,    7,    17,     31,     49,      71, ...
   0, 1,   26,    99,    244,    485,     846, ...
   1, 1,   97,   577,   1921,   4801,   10081, ...
   0, 1,  362,  3363,  15124,  47525,  120126, ...
  -1, 1, 1351, 19601, 119071, 470449, 1431431, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Mirror of A101124.
Columns 0-20 give A056594, A000012, A001075, A001541, A001091, A001079, A023038, A011943(n+1), A001081, A023039, A001085, A077422, A077424, A097308, A097310, A068203, A322888, A056771, A322889, A078986, A322890.
Rows 0-10 give A000012, A001477, A056220, A144129, A144130, A243131, A243132, A243133, A243134, A243135, A243136.
Main diagonal gives A115066.
Cf. A323182 (Chebyshev polynomial of the second kind).

Table[ChebyshevT[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 28 2018 *)

T(n,k) = polchebyshev(n,1,k);
matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Dec 28 2018

T(n, k) = round(cos(n*acos(k)));\\ Seiichi Manyama, Mar 05 2021

T(n, k) = if(n==0, 1, n*sum(j=0, n, (2*k-2)^j*binomial(n+j, 2*j)/(n+j))); \\ Seiichi Manyama, Mar 05 2021

A(0,k) = 1, A(1,k) = k and A(n,k) = 2 * k * A(n-1,k) - A(n-2,k) for n > 1.
A(n,k) = cos(n*arccos(k)). - Seiichi Manyama, Mar 05 2021
A(n,k) = n * Sum_{j=0..n} (2*k-2)^j * binomial(n+j,2*j)/(n+j) for n > 0. - Seiichi Manyama, Mar 05 2021

A342205 a(n) = T(n,n+1) where T(n,x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 2, 17, 244, 4801, 120126, 3650401, 130576328, 5374978561, 250283080090, 13007560326001, 746411226303612, 46873096812360001, 3197490648645613334, 235451028081583642049, 18614381236112230383376, 1572584048032918633353217

Offset: 0

Seiichi Manyama, Mar 05 2021

Seiichi Manyama, Table of n, a(n) for n = 0..351
Wikipedia, Chebyshev polynomials.

Cf. A115066, A323117, A342206, A342207, A370259, A370260, A370261.

Table[ChebyshevT[n, n + 1], {n, 0, 16}] (* Amiram Eldar, Mar 05 2021 *)

PARI
```
a(n) = polchebyshev(n, 1, n+1);
```
PARI
```
a(n) = round(cos(n*acos(n+1)));
```

a(n) = if(n==0, 1, n*sum(k=0, n, (2*n)^k*binomial(n+k, 2*k)/(n+k)));

a(n) = cos(n*arccos(n+1)).
a(n) = n * Sum_{k = 0..n} (2*n)^k * binomial(n+k,2*k)/(n+k) for n > 0.
From Peter Bala, Mar 11 2024: (Start)
a(2*n+1) == 1 (mod (2*n + 1)^3); a(2*n) == 1 (mod (n + 1)*(2*n)^3).
a(n) = hypergeom([n, -n], [1/2], -n/2). (End)
a(n) ~ exp(1) * 2^(n-1) * n^n. - Vaclav Kotesovec, Mar 12 2024

A101124 Number triangle associated to Chebyshev polynomials of first kind.

Original entry on oeis.org

1, 0, 1, -1, 1, 1, 0, 1, 2, 1, 1, 1, 7, 3, 1, 0, 1, 26, 17, 4, 1, -1, 1, 97, 99, 31, 5, 1, 0, 1, 362, 577, 244, 49, 6, 1, 1, 1, 1351, 3363, 1921, 485, 71, 7, 1, 0, 1, 5042, 19601, 15124, 4801, 846, 97, 8, 1, -1, 1, 18817, 114243, 119071, 47525, 10081, 1351, 127, 9, 1, 0, 1, 70226, 665857, 937444, 470449, 120126, 18817, 2024, 161

Offset: 0

Paul Barry, Dec 02 2004

			As a number triangle, rows begin:
  {1},
  {0,1},
  {-1,1,1},
  {0,1,2,1},
  ...
As a square array, rows begin
   1, 1,  1,   1,    1, ...
   0, 1,  2,   3,    4, ...
  -1, 1,  7,  17,   31, ...
   0, 1, 26,  99,  244, ...
   1, 1, 97, 577, 1921, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns include A001075, A001541, A001091, A001079, A023038, A011943.
Row sums are A101125.
Diagonal sums are A101126.
Main diagonal gives A115066.
Mirror of A322836.
Cf. A053120.

T[n_, k_] := SeriesCoefficient[x^k (1 - k x)/(1 - 2 k x + x^2), {x, 0, n}];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 12 2017 *)

Number triangle S(n, k)=T(n-k, k), k

Columns have g.f. x^k(1-kx)/(1-2kx+x^2).

Also, square array if(n=0, 1, T(n, k)) read by antidiagonals.

A349071 a(n) = T(n, 2*n), where T(n, x) is the Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 2, 31, 846, 32257, 1580050, 94558751, 6686381534, 545471324161, 50428155189474, 5210183616019999, 594949288292777902, 74404881332329766401, 10114032809617941274226, 1484781814660796486716447, 234114571438498509048719550, 39459584112457284328544403457

Offset: 0

Vaclav Kotesovec, Nov 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..321
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind.
Wikipedia, Chebyshev polynomials.

Cf. A053120, A115066, A173129, A349072, A349073.

Mathematica
```
Table[ChebyshevT[n, 2*n], {n, 0, 20}]
```

a(n) = polchebyshev(n, 1, 2*n); \\ Michel Marcus, Nov 07 2021

from sympy import chebyshevt
def A349071(n): return chebyshevt(n,n<<1) # Chai Wah Wu, Nov 08 2023

a(n) = cosh(n*arccosh(2*n)).
a(n) = ((2*n + sqrt(4*n^2-1))^n + (2*n - sqrt(4*n^2-1))^n)/2.
a(n) ~ 2^(2*n-1) * n^n.

A349070 a(n) = T(3*n, n), where T(n, x) is the Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 1, 1351, 3880899, 28355806081, 429364731169925, 11731978174095849671, 525735133485219615486151, 36049049892983023583045990401, 3588952618789973294871796462342089, 497937643558017209960022199517744044999, 93156956377055671178035977181412016527566091

Offset: 0

Vaclav Kotesovec, Nov 07 2021

nonn

In general, for k>=1, T(k*n, n) ~ 2^(k*n - 1) * n^(k*n).

Seiichi Manyama, Table of n, a(n) for n = 0..136
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind.
Wikipedia, Chebyshev polynomials.

Cf. A053120, A115066, A173129, A349072, A349074.

Mathematica
```
Table[ChebyshevT[3*n, n], {n, 0, 13}]
```

a(n) = polchebyshev(3*n, 1, n); \\ Michel Marcus, Nov 07 2021

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

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

Vaclav Kotesovec, Nov 07 2021

nonn

In general, for k>=1, T(n, k*n) ~ 2^(n-1) * k^n * n^n.

Seiichi Manyama, Table of n, a(n) for n = 0..306
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind.
Wikipedia, Chebyshev polynomials.

Cf. A053120, A115066, A349070, A349071, A349076.

Mathematica
```
Table[ChebyshevT[n, 3*n], {n, 0, 20}]
```

a(n) = polchebyshev(n, 1, 3*n); \\ Michel Marcus, Nov 07 2021

from sympy import chebyshevt
def A349072(n): return chebyshevt(n,3*n) # Chai Wah Wu, Nov 08 2023

a(n) = cosh(n*arccosh(3*n)).
a(n) = ((3*n + sqrt(9*n^2-1))^n + (3*n - sqrt(9*n^2-1))^n)/2.
a(n) ~ 2^(n-1) * 3^n * n^n.

A323117 a(n) = T_{n}(n-1) where T_{n}(x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 0, 1, 26, 577, 15124, 470449, 17057046, 708158977, 33165873224, 1730726404001, 99612037019890, 6269617090376641, 428438743526336412, 31592397706723526737, 2500433598371461203374, 211434761022028192051201, 19023879409608991280267536

Offset: 0

Seiichi Manyama, Jan 05 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..351
Wikipedia, Chebyshev polynomials.

Cf. A115066, A322836, A323118.

Table[ChebyshevT[n, n - 1], {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)

PARI
```
a(n) = polchebyshev(n, 1, n-1);
```

a(n) = round(cos(n*acos(n-1))); \\ Seiichi Manyama, Mar 05 2021

a(n) = if(n==0, 1, n*sum(k=0, n, (2*n-4)^k*binomial(n+k, 2*k)/(n+k))); \\ Seiichi Manyama, Mar 05 2021

a(n)^2 - ((n - 1)^2 - 1) * A323118(n-1)^2 = 1 for n > 0.
a(n) = A322836(n,n-1) for n > 0.
a(n) ~ exp(-1) * 2^(n-1) * n^n. - Vaclav Kotesovec, Jan 05 2019
a(n) = cos(n*arccos(n-1)). - Seiichi Manyama, Mar 05 2021
a(n) = n * Sum_{k=0..n} (2*n-4)^k * binomial(n+k,2*k)/(n+k) for n > 0. - Seiichi Manyama, Mar 05 2021

A343259 a(n) = 2 * T(n,n/2) where T(n,x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

2, 1, 2, 18, 194, 2525, 39202, 710647, 14760962, 345946302, 9034502498, 260219353691, 8195978831042, 280256592535933, 10340256951198914, 409468947059131650, 17322711762013765634, 779742677038695037937, 37210469265847998489922, 1876572071974094803391179

Offset: 0

Seiichi Manyama, Apr 09 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..386
Wikipedia, Chebyshev polynomials.

Main diagonal of A298675.

Cf. A097690, A115066, A342205, A343260, A343261.

Table[2*ChebyshevT[n, n/2], {n, 1, 20}] (* Amiram Eldar, Apr 09 2021 *)

PARI
```
a(n) = 2*polchebyshev(n, 1, n/2);
```
PARI
```
a(n) = round(2*cos(n*acos(n/2)));
```

a(n) = if(n==0, 2, 2*n*sum(k=0, n, (n-2)^k*binomial(n+k, 2*k)/(n+k)));

a(n) = 2 * cos(n*arccos(n/2)).
a(n) = 2 * n * Sum_{k=0..n} (n-2)^k * binomial(n+k,2*k)/(n+k) for n > 0.
a(n) ~ n^n. - Vaclav Kotesovec, Apr 09 2021

Showing 1-10 of 13 results. Next

cp's OEIS Frontend

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

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A323118 a(n) = U_{n}(n) where U_{n}(x) is a Chebyshev polynomial of the second kind.

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A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

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A342205 a(n) = T(n,n+1) where T(n,x) is a Chebyshev polynomial of the first kind.

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A101124 Number triangle associated to Chebyshev polynomials of first kind.

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A349071 a(n) = T(n, 2*n), where T(n, x) is the Chebyshev polynomial of the first kind.

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A349070 a(n) = T(3*n, n), where T(n, x) is the Chebyshev polynomial of the first kind.

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