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.

A128055 a(n) = 2^A128054(n).

Original entry on oeis.org

1, 1, 2, 4, 8, 32, 64, 64, 128, 256, 512, 2048, 4096, 4096, 8192, 16384, 32768, 131072, 262144, 262144, 524288, 1048576, 2097152, 8388608, 16777216, 16777216, 33554432, 67108864, 134217728, 536870912, 1073741824
Offset: 0

Views

Author

Paul Barry, Feb 13 2007

Keywords

Comments

A factor in A128056.
The signed sequence 1,1,2,-4,-8,-32,-64,-64,-128,256,512... is the Hankel transform of the doubled sequence 1,1,2,2,6,6,... of central binomial coefficients (A128014). - Paul Barry, Sep 09 2008

Links

Crossrefs

Cf. A128014, A128054, A128056.

Programs

  • Mathematica
    LinearRecurrence[{2,-4,8,-16,32},{1,1,2,4,8},31] (* James C. McMahon, Jan 05 2025 *)

Formula

G.f.: (-1-16*x^4+4*x^3-4*x^2+x)/((2*x-1)*(4*x^2-2*x+1)*(4*x^2+2*x+1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
Sum_{n>=0} 1/a(n) = 62/21. - Amiram Eldar, Mar 28 2023

A128056 Hankel transform of A128057.

Original entry on oeis.org

1, -3, -6, 28, 56, -288, -576, 3008, 6016, -31488, -62976, 329728, 659456, -3452928, -6905856, 36159488, 72318976, -378667008, -757334016, 3965452288, 7930904576, -41526755328, -83053510656, 434873827328
Offset: 0

Views

Author

Paul Barry, Feb 13 2007

Keywords

Comments

a(n)=2^A128054(n)*A128053(n). Hankel transform of A128058.

Formula

a(n)=A128055(n)*((cos(pi*n/2)-sin(pi*n/2))((F(n-1)+F(n+1))(5/6-cos(pi*n/3)/3)(1+(-1)^n)/2 +(F(n)+F(n+2))(5/6-cos(pi*(n+1)/3)/3)(1-(-1)^n)/2)).
Empirical g.f.: -(2*x-1)*(4*x^2-x+1) / (16*x^4+12*x^2+1). - Colin Barker, Jun 27 2013

A162533 a(n) = Sum_{k=0..n} binomial(n,2k)*A002426(k).

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 68, 176, 454, 1174, 3052, 7976, 20932, 55108, 145448, 384704, 1019462, 2706214, 7194956, 19155896, 51065260, 136284236, 364097912, 973654240, 2605983772, 6980545276, 18712478072, 50196568144, 134739960904, 361892443592, 972537193168
Offset: 0

Views

Author

Paul Barry, Jul 05 2009

Keywords

Comments

Hankel transform is (-1)^binomial(n,2)*(-2)^A128054(n) (see A128055).

Links

Crossrefs

Cf. A027826.

Programs

  • Mathematica
    b[n_] := If[n < 0, 0, 3^n Hypergeometric2F1[1/2, -n, 1, 4/3]]; Table[Sum[Binomial[n, 2*k]*b[k], {k, 0, n}], {n, 0, 50}] (* or *) CoefficientList[Series[(1-x)/sqrt(1-4*x+4*x^2-4*x^4), {x, 0, 50}], x] (* G. C. Greubel, Feb 27 2017 *)
  • PARI
    x='x+O('x^50); Vec((1-x)/sqrt(1-4*x+4*x^2-4*x^4)) \\ G. C. Greubel, Feb 27 2017

Formula

G.f.: (1-x)/((1-x)^2-x^2-2x^4/((1-x)^2-x^2-x^4/((1-x)^2-x^2-x^4/(1-... (continued fraction).
G.f.: (1-x)/sqrt(1-4*x+4*x^2-4*x^4) = (1-x)/sqrt((1-2*x)^2-4*x^4) = (1-x)/sqrt((1-x-2*x^2)*(1-x+2*x^2)). - Paul Barry, Oct 13 2009
Conjecture: n*a(n) + (4-5*n)*a(n-1) + 2*(4*n-7)*a(n-2) + 4*(3-n)*a(n-3) + 4*(2-n)*a(n-4) + 4*(n-4)*a(n-5) = 0. - R. J. Mathar, Nov 16 2011
a(n) ~ 3^(1/4) * (1 + sqrt(3))^(n + 1/2) / (2^(3/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jun 08 2019
Showing 1-3 of 3 results.

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