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A105696 Expansion of (1-x)/sqrt((1-3*x)/(1+x)).

%I A105696 #16 Nov 19 2021 08:48:45
%S A105696 1,1,2,6,16,44,122,342,966,2746,7846,22514,64836,187288,542438,
%T A105696 1574666,4580400,13347324,38956182,113861922,333226560,976353876,
%U A105696 2863756158,8407877394,24707200854,72663608178,213864889770,629893319902
%N A105696 Expansion of (1-x)/sqrt((1-3*x)/(1+x)).
%C A105696 Apply the Riordan array (1-x,x/(1+x)) to C(2n,n).
%H A105696 G. C. Greubel, <a href="/A105696/b105696.txt">Table of n, a(n) for n = 0..1000</a>
%F A105696 a(n) = A002426(n) - A002426(n-2).
%F A105696 a(n) = Sum_{k=0..floor(n/2)} ( 2*C(n-2, k)*C(n-k, k) ) - C(1, n).
%F A105696 Conjecture D-finite with recurrence: n*a(n) +(-3*n+2)*a(n-1) +(-n+2)*a(n-2) +3*(n-4)*a(n-3)=0. - _R. J. Mathar_, Jan 22 2020
%F A105696 a(n) ~ 4 * 3^(n - 3/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Nov 19 2021
%t A105696 CoefficientList[Series[(1-x)/Sqrt[(1-3x)/(1+x)],{x,0,30}],x] (* _Harvey P. Dale_, Feb 12 2016 *)
%o A105696 (PARI) x='x+O('x^50); Vec((1-x)/sqrt((1-3*x)/(1+x))) \\ _G. C. Greubel_, Mar 02 2017
%Y A105696 Cf. A002426.
%K A105696 easy,nonn
%O A105696 0,3
%A A105696 _Paul Barry_, Apr 17 2005