A349076 -id:A349076 - 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

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.

A349074 a(n) = U(3*n, n), where U(n, x) is the Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 4, 2911, 7997214, 57641556673, 867583274883920, 23630375698358890319, 1056918444955456528983706, 72383076947075470731692782081, 7200266529428094485775774835670652, 998383804974887102441600687728515247999, 186701261436825568741051032736345268517903734

Offset: 0

Vaclav Kotesovec, Nov 07 2021

nonn

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

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

Cf. A323118, A349070, A349073, A349076.

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

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

from sympy import chebyshevu
def A349074(n): return chebyshevu(3*n,n) # Chai Wah Wu, Nov 08 2023

For n>1, a(n) = ((n + sqrt(n^2-1))^(3*n+1) - (n - sqrt(n^2-1))^(3*n+1)) / (2*sqrt(n^2-1)).

a(n) ~ 2^(3*n) * n^(3*n).

A349075 a(n) = U(n, 2*n), where U(n, x) is the Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 4, 63, 1704, 64769, 3168060, 189447551, 13389885712, 1092011153409, 100934312212404, 10426892198423999, 1190514147664125240, 148874434455514989313, 20235554722675691942764, 2970511463324707397138175, 468359315014627272862943520, 78938449723310515780367269889

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 Second Kind.
Wikipedia, Chebyshev polynomials.

Cf. A323118, A349073, A349076.

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

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

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

a(n) ~ 4^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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A349072 a(n) = T(n, 3*n), where T(n, x) is the Chebyshev polynomial of the first kind.

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Programs

Mathematica

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Formula

A349074 a(n) = U(3*n, n), where U(n, x) is the Chebyshev polynomial of the second kind.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A349075 a(n) = U(n, 2*n), where U(n, x) is the Chebyshev polynomial of the second kind.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula