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%I #16 Mar 14 2019 17:09:45
%S 1,3,10,36,144,648,3384,20520,145728,1181952,10917504,111601152,
%T 1265777280,15544566144,208320719616,2980582728192,46020833427456,
%U 751100760576000,13121167636058112,240473024248393728,4687531209011183616,95293672221284794368
%N Expansion of exp(3x)/sqrt(1-x^2).
%C Binomial transform of A081920.
%H Robert Israel, <a href="/A081921/b081921.txt">Table of n, a(n) for n = 0..449</a>
%F E.g.f. exp(3x)/sqrt(1-x^2)
%F a(n) = 3^n*n!*Sum_{k=0..floor(n/2)} binomial(2*k, k)/(n-2*k)!/36^k. - _Vladeta Jovovic_, Oct 11 2003
%F Conjecture: a(n)-3*a(n-1) -(n-1)^2*a(n-2) +3*(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Nov 24 2012
%F a(n) ~ (exp(6)+(-1)^n)*n^n/exp(n+3). - _Vaclav Kotesovec_, Oct 05 2013
%F Mathar's conjecture follows from the differential equation (-3*x^2+x+3)*y+(x^2-1)*y'=0 satisfied by the E.g.f. - _Robert Israel_, Mar 14 2019
%p f:= gfun:-rectoproc({a(n)-3*a(n-1)-(n-1)^2*a(n-2)+(3*(n-1))*(n-2)*a(n-3),a(0)=1,a(1)=3,a(2)=10},a(n),remember):
%p map(f, [$0..30]); # _Robert Israel_, Mar 14 2019
%t CoefficientList[Series[E^(3*x)/Sqrt[1-x^2], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 05 2013 *)
%Y Cf. A081922.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 01 2003
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