A081920 - OEIS

%I #10 Feb 19 2018 22:03:35

%S 1,2,5,14,49,202,1069,6470,48353,391058,3767029,37936318,445650385,

%T 5359634906,74198053661,1036667808758,16516851030721,262805595346210,

%U 4735033850606437,84510767762583662,1698609728377283441

%N Expansion of exp(2x)/sqrt(1-x^2).

%C Binomial transform of A081919

%H Robert Israel, <a href="/A081920/b081920.txt">Table of n, a(n) for n = 0..449</a>

%F E.g.f. exp(2x)/sqrt(1-x^2).

%F Conjecture: a(n) -2*a(n-1) -(n-1)^2*a(n-2) +2*(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Nov 24 2012

%F Conjecture confirmed using d.e. (x^2-1)*y' + (-2*x^2+x+2)*y = 0 satisfied by the E.g.f. - _Robert Israel_, Feb 19 2018

%F a(n) ~ n^n * (exp(2)+(-1)^n*exp(-2)) / exp(n). - _Vaclav Kotesovec_, Feb 04 2014

%p f:= gfun:-rectoproc({a(n) -2*a(n-1) -(n-1)^2*a(n-2) +2*(n-1)*(n-2)*a(n-3)=0, a(0)=1,a(1)=2,a(2)=5},a(n),remember):

%p map(f, [$0..30]); # _Robert Israel_, Feb 19 2018

%t CoefficientList[Series[E^(2*x)/Sqrt[1-x^2], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Feb 04 2014 *)

%Y Cf. A081921.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Apr 01 2003