A158495 - OEIS

%I #20 Jan 10 2023 07:59:58

%S 1,-1,-3,-8,-21,-56,-154,-440,-1309,-4048,-12958,-42712,-144210,

%T -496432,-1735764,-6145968,-21986781,-79331232,-288307254,-1054253208,

%U -3875769606,-14315659632,-53097586284,-197677736208,-738415086066

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

%C Hankel transform is A158496.

%H G. C. Greubel, <a href="/A158495/b158495.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (2*0^n - 2^n + A126966(n))/2.

%F 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 From _G. C. Greubel_, Jan 09 2023: (Start)

%F a(n) = [n=0] - Sum_{k=1..n} 2^(n-k)*A000108(k-1).

%F a(n) = Sum_{j=0..n} 2^(n-j)*A246432(j). (End)

%t CoefficientList[Series[((1-4x)+Sqrt[1-4x])/(2(1-2x)),{x,0,30}],x] (* _Harvey P. Dale_, Dec 15 2011 *)

%o (Magma)

%o A158495:= func< n | n eq 0 select 1 else - (&+[2^(n-j)*Catalan(j-1): j in [1..n]]) >;

%o [A158495(n): n in [0..40]]; // _G. C. Greubel_, Jan 09 2023

%o (SageMath)

%o def A158495(n): return int(n==0) - sum(2^(n-k)*catalan_number(k-1) for k in range(1,n+1))

%o [A158495(n) for n in range(41)] # _G. C. Greubel_, Jan 09 2023

%Y Essentially the same as A014318, up to sign and offset.

%Y Cf. A126966, A158496, A246432.

%K easy,sign

%O 0,3

%A _Paul Barry_, Mar 20 2009