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

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

Author

Paul Barry, Mar 28 2003

Keywords

• easy
• nonn

Comments

Inverse binomial transform of A081673.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A000984, A081673.

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