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A158495 Expansion of ((1-4*x) + sqrt(1-4*x))/(2*(1-2*x)).

%I A158495 #20 Jan 10 2023 07:59:58
%S A158495 1,-1,-3,-8,-21,-56,-154,-440,-1309,-4048,-12958,-42712,-144210,
%T A158495 -496432,-1735764,-6145968,-21986781,-79331232,-288307254,-1054253208,
%U A158495 -3875769606,-14315659632,-53097586284,-197677736208,-738415086066
%N A158495 Expansion of ((1-4*x) + sqrt(1-4*x))/(2*(1-2*x)).
%C A158495 Hankel transform is A158496.
%H A158495 G. C. Greubel, <a href="/A158495/b158495.txt">Table of n, a(n) for n = 0..1000</a>
%F A158495 a(n) = (2*0^n - 2^n + A126966(n))/2.
%F A158495 Conjecture: n*a(n) +6*(-n+1)*a(n-1) +4*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Dec 03 2014
%F A158495 From _G. C. Greubel_, Jan 09 2023: (Start)
%F A158495 a(n) = [n=0] - Sum_{k=1..n} 2^(n-k)*A000108(k-1).
%F A158495 a(n) = Sum_{j=0..n} 2^(n-j)*A246432(j). (End)
%t A158495 CoefficientList[Series[((1-4x)+Sqrt[1-4x])/(2(1-2x)),{x,0,30}],x] (* _Harvey P. Dale_, Dec 15 2011 *)
%o A158495 (Magma)
%o A158495 A158495:= func< n | n eq 0 select 1 else - (&+[2^(n-j)*Catalan(j-1): j in [1..n]]) >;
%o A158495 [A158495(n): n in [0..40]]; // _G. C. Greubel_, Jan 09 2023
%o A158495 (SageMath)
%o A158495 def A158495(n): return int(n==0) - sum(2^(n-k)*catalan_number(k-1) for k in range(1,n+1))
%o A158495 [A158495(n) for n in range(41)] # _G. C. Greubel_, Jan 09 2023
%Y A158495 Essentially the same as A014318, up to sign and offset.
%Y A158495 Cf. A126966, A158496, A246432.
%K A158495 easy,sign
%O A158495 0,3
%A A158495 _Paul Barry_, Mar 20 2009