A349074 -id:A349074 - cp's OEIS frontend

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

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

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

Vaclav Kotesovec, Nov 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..193
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind.
Wikipedia, Chebyshev polynomials.

Cf. A173129, A323118, A349071, A349074, A349075.

Mathematica
```
Table[ChebyshevU[2*n, n], {n, 0, 15}]
```

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

from sympy import chebyshevu
def A349073(n): return chebyshevu(n<<1,n) # Chai Wah Wu, Nov 08 2023

For n>1, a(n) = ((n + sqrt(n^2-1))^(2*n+1) - (n - sqrt(n^2-1))^(2*n+1)) / (2*sqrt(n^2-1)).

a(n) ~ 2^(2*n) * n^(2*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).

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

Vaclav Kotesovec, Nov 07 2021

nonn

In general, for k>=1, U(n, k*n) ~ 2^n * 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 Second Kind.
Wikipedia, Chebyshev polynomials.

Cf. A323118, A349071, A349072, A349074, A349075.

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

a(n) = polchebyshev(n, 2, 3*n); \\ Michel Marcus, Nov 07 2021

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

a(n) ~ 6^n * n^n.

cp's OEIS Frontend

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

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A349073 a(n) = U(2*n, n), where U(n, x) is the Chebyshev polynomial of the second kind.

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Mathematica

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Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349076 a(n) = U(n, 3*n), where U(n, x) is the Chebyshev polynomial of the second kind.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula