This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A158499 #29 Mar 19 2025 10:16:12
%S A158499 1,1,1,0,-5,-24,-90,-312,-1053,-3536,-11934,-40664,-140114,-488240,
%T A158499 -1719380,-6113200,-21921245,-79200160,-288045110,-1053728920,
%U A158499 -3874721030,-14313562480,-53093391980,-197669347600,-738398308850,-2766700765024
%N A158499 Expansion of (1 + sqrt(1-4*x))/(2-4*x).
%C A158499 Hankel transform is A056594 with g.f. 1/(1+x^2).
%C A158499 Row sums of the Riordan array (sqrt(1-4*x)/(1-2*x), x*c(x)^2), c(x) the g.f. of A000108.
%C A158499 The inverse Catalan transform yields A146559. - _R. J. Mathar_, Mar 20 2009
%H A158499 Matthew House, <a href="/A158499/b158499.txt">Table of n, a(n) for n = 0..1669</a>
%H A158499 Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%F A158499 a(n) = Sum_{k=0..n} binomial(2*k,k)*A158495(n-k).
%F A158499 n*a(n) + 6*(1-n)*a(n-1) + 4*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Nov 14 2011
%t A158499 CoefficientList[ Series[(1 + Sqrt[1 - 4x])/(2 - 4x), {x, 0, 26}], x] (* _Robert G. Wilson v_, Nov 08 2015 *)
%o A158499 (PARI) my(x='x+O('x^33)); Vec(((1-4*x)+sqrt(1-4*x))/(2*(1-2*x)*sqrt(1-4*x))) \\ _Altug Alkan_, Nov 08 2015
%o A158499 (Magma) [n le 2 select 1 else (6*(n-2)*Self(n-1) - 4*(2*n-5)*Self(n-2))/(n-1): n in [1..30]]; // _G. C. Greubel_, Mar 17 2025
%o A158499 (SageMath)
%o A158499 @CachedFunction
%o A158499 def a(n): # a = A158499
%o A158499 if n<2: return 1
%o A158499 else: return (6*(n-1)*a(n-1) - 4*(2*n-3)*a(n-2))/n
%o A158499 [a(n) for n in range(41)] # _G. C. Greubel_, Mar 17 2025
%Y A158499 Cf. A000108, A056594, A146559, A158495.
%K A158499 easy,sign
%O A158499 0,5
%A A158499 _Paul Barry_, Mar 20 2009
%E A158499 Name edited by _Matthew House_, Nov 08 2015