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%I #10 Feb 01 2024 00:26:27
%S 1,1,6,18,70,246,924,3412,12870,48550,184756,705180,2704156,10399676,
%T 40116600,155114088,601080390,2333593350,9075135300,35345215180,
%U 137846528820,538257689684,2104098963720,8233430022168,32247603683100
%N Expansion of 1/sqrt(1-4*x) - x/sqrt(1-4*x^2).
%C Partial sums are A129368.
%H G. C. Greubel, <a href="/A129369/b129369.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = binomial(2*n, n) - (1/2)*(1-(-1)^n)*binomial(n-1, (n-1)/2).
%F From _G. C. Greubel_, Jan 31 2024: (Start)
%F a(2*n) = A001448(n).
%F a(2*n+1) = (n+1)*( 2*A000108(2*n+1) - A000108(n) ).
%F a(2*n+1) = 2*A002458(n) - A000984(n).
%F (n+1)^3*(n+2)*(n+3)*a(n+3) = 2*(n+1)*(n+2)^3*(2*n+1)*a(n+2) + 4*(n+1)^4*(n+3)*a(n+1) - 8*n*(n+2)^3*(2*n+1)*a(n), with a(0)=a(1) = 1, a(2) = 6. (End)
%t CoefficientList[Series[1/Sqrt[1-4x]-x/Sqrt[1-4x^2],{x,0,30}],x] (* _Harvey P. Dale_, Feb 02 2012 *)
%o (Magma) B:=Binomial; [B(2*n,n) - (n mod 2)*B(n-1, Floor((n-1)/2)): n in [0..60]]; // _G. C. Greubel_, Jan 31 2024
%o (SageMath) [binomial(2*n,n) - (n%2)*binomial(n-1, (n-1)//2) for n in range(61)] # _G. C. Greubel_, Jan 31 2024
%Y Cf. A000108, A000984, A001448, A002458, A129368.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Apr 11 2007
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