A081669 - OEIS

%I #9 Feb 13 2014 02:29:06

%S 1,3,7,15,35,83,205,521,1363,3651,9977,27701,77885,221133,632611,

%T 1820375,5262163,15266003,44414953,129521141,378427945,1107447881,

%U 3245329831,9521616731,27965113597,82210390733,241880335015

%N Expansion of exp(2x)+exp(x)BesselI_0(2x).

%C Binomial transform of A081668. Inverse binomial transform of A081670.

%H Vincenzo Librandi, <a href="/A081669/b081669.txt">Table of n, a(n) for n = 0..1000</a>

%F E.g.f. exp(2x) - exp(0) + exp(x)BesselI_0(2x).

%F Conjecture: n*a(n) +(-6*n+5)*a(n-1) +(9*n-17)*a(n-2) +4*(n-1)*a(n-3) +12*(-n+3)*a(n-4)=0. - _R. J. Mathar_, Nov 24 2012

%F Recurrence: n*(3*n-8)*a(n) = (3*n-2)*(4*n-9)*a(n-1) - (3*n^2 - 5*n + 6)*a(n-2) - 6*(n-2)*(3*n-5)*a(n-3). - _Vaclav Kotesovec_, Feb 12 2014

%F a(n) ~ 3^(n+1/2) / (2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 12 2014

%t CoefficientList[Series[E^(2*x) - 1 + E^x*BesselI[0,2*x],{x,0,20}],x]*Range[0,20]! (* _Vaclav Kotesovec_, Feb 12 2014 *)

%Y Cf. A000984.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 28 2003