A081673 Expansion of exp(3*x) - exp(x)*(1-BesselI_0(2*x)).

1, 3, 11, 33, 99, 293, 869, 2579, 7667, 22821, 68001, 202799, 605229, 1807263, 5399195, 16136513, 48243347, 144275093, 431573297, 1291258319, 3864163769, 11565703931, 34622195135, 103656406949, 310377872861, 929465445743

Offset: 0

Author

Paul Barry, Mar 28 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A081672.

Links

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

Crossrefs

Cf. A025191, A027914, A027915.

Programs

Maple

Egf:= exp(3*x)-exp(x)*(1-BesselI(0,2*x)):
S:= series(Egf,x,101):
seq(coeff(S,x,n)*n!, n=0..100); # Robert Israel, Jun 03 2016

Mathematica

CoefficientList[Series[E^(3*x)-E^x*(1-BesselI[0,2*x]), {x, 0, 50}], x] * Range[0, 50]! (* Vaclav Kotesovec, Jul 02 2015 *)

Formula

E.g.f. exp(3*x) - exp(x)*(1-BesselI_0(2*x)).

Conjecture: n*(2*n - 7)*a(n) +(-12*n^2 + 50*n - 33)*a(n-1) +(16*n^2 - 76*n + 87)*a(n-2) +3*(4*n^2 - 22*n + 27)*a(n-3) -9*(2*n-5)*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 24 2012

a(n) ~ 3^n * (1 + sqrt(3)/(2*sqrt(Pi*n))). - Vaclav Kotesovec, Jul 02 2015

From Robert Israel, Jun 03 2016: (Start)

E.g.f. A(x) satisfies

-27 A + (-63 x + 9) A' + (-18 x^2 + 42 x + 39) A'' + (12 x^2 + 68 x - 25) A''' + (16 x^2 - 58 x + 4) A'''' + (-12 x^2 + 11 x) A''''' + 2 x^2 A'''''' = 0.

This implies Mathar's conjectured recursion. (End)