A323118 -id:A323118 - cp's OEIS frontend

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

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.

A318192 a(n) = U_{n}(n)/(n+1) where U_{n}(x) is a Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 1, 5, 51, 781, 16005, 411881, 12776743, 464278585, 19350109449, 910126036909, 47694593157211, 2755988277318277, 174100457124362509, 11937317942278298961, 882942450221936166735, 70077737629245663437041, 5940877531422707027770385

Offset: 0

Seiichi Manyama, Jan 07 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..352
Wikipedia, Chebyshev polynomials.

Cf. A323118.

PARI
```
{a(n) = polchebyshev(n, 2, n)/(n+1)}
```

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

a(n) = A323118(n)/(n+1).

a(n) = Sum_{k=0..floor(n/2)} (1/(2*k+1)) * binomial(n,2*k)*(n^2-1)^k*n^(n-2*k).

A342207 a(n) = U(n,n+1) where U(n,x) is a Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 4, 35, 496, 9701, 241956, 7338631, 262184896, 10783446409, 501827040100, 26069206375211, 1495427735314800, 93885489910449901, 6403169506981578436, 471427031236487965199, 37265225545829174607616, 3147895910861898495432209

Offset: 0

Seiichi Manyama, Mar 05 2021

nonn

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

Cf. A107995, A323118, A342168, A342205.

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

PARI
```
a(n) = polchebyshev(n, 2, n+1);
```

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

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

a(n) = Sum_{k=0..n} (2*n)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (2*n)^k * binomial(n+1+k,2*k+1).

a(n) ~ exp(1) * 2^n * n^n. - Vaclav Kotesovec, Mar 05 2021

cp's OEIS Frontend

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

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A318192 a(n) = U_{n}(n)/(n+1) where U_{n}(x) is a Chebyshev polynomial of the second kind.

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Author

Keywords

Links

Crossrefs

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PARI

PARI

Formula

A342207 a(n) = U(n,n+1) where U(n,x) is a Chebyshev polynomial of the second kind.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula