cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Seiichi Manyama, Jan 05 2019

Keywords

Links

Crossrefs

Main diagonal of A323182.
Cf. A115066, A318192, A323117, A349073, A349074, A349075, A349076.

Programs

  • Mathematica
    Table[ChebyshevU[n, n], {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)
  • PARI
    a(n) = polchebyshev(n, 2, n);
  • PARI
    a(n) = sum(k=0, n\2, (n^2-1)^k*n^(n-2*k)*binomial(n+1, 2*k+1));
  • PARI
    a(n) = sum(k=0, n, (2*n-2)^k*binomial(n+1+k, 2*k+1)); \\ Seiichi Manyama, Mar 03 2021

Formula

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

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

Views

Author

Seiichi Manyama, Mar 05 2021

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

A342206 a(n) = T(n,n+2) where T(n,x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 3, 31, 485, 10081, 262087, 8193151, 299537289, 12545596801, 592479412811, 31154649926687, 1805486216133613, 114342125644787041, 7857107443850071695, 582268591681887560191, 46292552162781456490001, 3930448770533424343942657
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2021

Keywords

Links

Crossrefs

Cf. A107995, A115066, A323117, A342205.

Programs

  • Mathematica
    Table[ChebyshevT[n, n + 2], {n, 0, 16}] (* Amiram Eldar, Mar 05 2021 *)
  • PARI
    a(n) = polchebyshev(n, 1, n+2);
  • PARI
    a(n) = round(cos(n*acos(n+2)));
  • PARI
    a(n) = if(n==0, 1, n*sum(k=0, n, (2*n+2)^k*binomial(n+k, 2*k)/(n+k)));

Formula

a(n) = cos(n*arccos(n+2)).
a(n) = n * Sum_{k=0..n} (2*n+2)^k * binomial(n+k,2*k)/(n+k) for n > 0.
a(n) ~ exp(2) * 2^(n-1) * n^n. - Vaclav Kotesovec, Mar 12 2024
Showing 1-3 of 3 results.

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