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.

A081672 Expansion of exp(2x) - exp(0) + BesselI_0(2x).

Original entry on oeis.org

1, 2, 6, 8, 22, 32, 84, 128, 326, 512, 1276, 2048, 5020, 8192, 19816, 32768, 78406, 131072, 310764, 524288, 1233332, 2097152, 4899736, 8388608, 19481372, 33554432, 77509464, 134217728, 308552056, 536870912, 1228859344
Offset: 0

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Author

Paul Barry, Mar 28 2003

Keywords

Comments

Inverse binomial transform of A081673.

Crossrefs

Programs

  • Maple
    1, seq(op([2^(2*k-1), 2^(2*k)+(2*k)!/k!^2]), k=1..30); # Robert Israel, Jun 03 2016
  • Mathematica
    CoefficientList[Series[1/Sqrt[1 - 4 z^2] + 1/(1 - 2 z) - 1, {z, 0, 20}], z] (* Benedict W. J. Irwin, Jun 03 2016 *)
    CoefficientList[Series[Exp[2*x] - 1 + BesselI[0, 2*x], {x, 0, 50}],
      x]*Range[0, 50]! (* G. C. Greubel, Jun 03 2016 *)
  • PARI
    a(n)=if(n,if(n%2,1,1+n!/(2^n*(n/2)!^2))<Charles R Greathouse IV, Jun 10 2016
    
  • PARI
    Vec(1/sqrt(1-4*x^2)+1/(1-2*x)-1) \\ Charles R Greathouse IV, Jun 10 2016

Formula

E.g.f.: exp(2x) - exp(0) + BesselI_0(2x).
Conjecture: n*a(n) +2*(1-n)*a(n-1) +4*(1-n)*a(n-2) +8*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 12 2012
a(n) ~ 2^n * (1+(1+(-1)^n)/sqrt(2*Pi*n)). - Vaclav Kotesovec, Feb 04 2014
From Benedict W. J. Irwin, Jun 03 2016: (Start)
For odd n, a(n) = 2^n. For even n>0, a(n) = 2^n*(1+n!/(2^n*(n/2)!^2)).
G.f.: 1/sqrt(1-4*z^2) + 1/(1-2*z) - 1. (End)
E.g.f. satisfies y''' - (2*x-2)*y'' - (4*x + 2)*y' + (8*x-4)*y + 8x - 4 = 0, which implies Mathar's conjectured recurrence. - Robert Israel, Jun 03 2016