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%I #16 Nov 13 2017 08:22:10
%S 1,3,63,3225,297675,42805665,8790957945,2433297161295,870928551367875,
%T 390718610250593625,214426984078881899325,141173178618822867992475,
%U 109729771971447612972712725,99352716603692210781106359375
%N E.g.f.: (1-x^2)^(-1/2)*exp(x^2/(1-x^2))*BesselI(0,x^2/(x^2-1)) (since this is an even function, we do not give the intercalating 0's).
%H Robert Israel, <a href="/A123687/b123687.txt">Table of n, a(n) for n = 0..221</a>
%F (n+1)*(2*n+3)^2*(2*n+1)^2*a(n) - (2*n+5)*(2*n+3)^2*a(n+1) + (n+2)*a(n+2) = 0. - _Robert Israel_, Oct 10 2016
%F a(n) ~ 2^(2*n - 1/4) * exp(2*sqrt(2*n) - 2*n - 1) * n^(2*n - 1/4) / sqrt(Pi) * (1 + 67/(48*sqrt(2*n))). - _Vaclav Kotesovec_, Nov 13 2017
%p G:=(1-x^2)^(-1/2)*exp(x^2/(1-x^2))*BesselI(0,x^2/(x^2-1)): Gser:=series(G,x=0,40): seq((2*n)!*coeff(Gser,x,2*n),n=0..15); # _Emeric Deutsch_, Oct 31 2006
%t DeleteCases[Flatten@ MapIndexed[#1 (#2 - 1)! &, CoefficientList[Series[(1 - x^2)^(-1/2) Exp[x^2/(1 - x^2)] BesselI[0, x^2/(x^2 - 1)], {x, 0, 26}], x]], 0] (* _Michael De Vlieger_, Oct 10 2016 *)
%t With[{nmax = 50}, CoefficientList[Series[(1 - x^2)^(-1/2)*Exp[x^2/(1 - x^2)]*BesselI[0, x^2/(x^2 - 1)], {x, 0, nmax}], x]*Range[0, nmax]!][[;; ;; 2 ]] (* _G. C. Greubel_, Oct 18 2017 *)
%Y Cf. A123510, A123511, A123512, A123525, A123686.
%K nonn
%O 0,2
%A _Karol A. Penson_, Oct 06 2006
%E More terms from _Emeric Deutsch_, Oct 31 2006
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