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 A100066 #46 Mar 18 2025 03:07:47
%S A100066 0,1,1,3,3,9,9,29,29,99,99,351,351,1275,1275,4707,4707,17577,17577,
%T A100066 66197,66197,250953,250953,956385,956385,3660541,3660541,14061141,
%U A100066 14061141,54177741,54177741,209295261,209295261,810375651,810375651
%N A100066 Expansion of x/((1-x)*sqrt(1-4*x^2)).
%C A100066 An inverse Chebyshev transform of x/(1-x+x^2), where the Chebyshev transform of g(x) is ((1-x^2)/(1+x^2))*g(x/(1+x^2)) and the inverse transform maps a g.f. A(x) to (1/sqrt(1-4*x^2))*A(x*c(x^2)) where c(x) is the g.f. of the Catalan numbers A000108.
%C A100066 Hankel transform of a(n+1) is A120582. The Hankel transform of a(n) is (-1)*[x^n] x/(1+2*x-4*x^2). - _Paul Barry_, Mar 29 2010
%H A100066 G. C. Greubel, <a href="/A100066/b100066.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi).
%F A100066 a(n) = Sum_{k=0..n} if(mod(n-k, 2)=0, binomial(n, (n-k)/2) * 2*sin(Pi*k/3) / sqrt(3)).
%F A100066 a(n) = Sum_{k=0..n} binomial(n, (n-k)/2)*(1+(-1)^(n-k))*sin(Pi*k/3)/sqrt(3).
%F A100066 a(n) = Sum_{k=0..n} binomial(n-1, (n-1)/2)*(1-(-1)^k)/2.
%F A100066 a(n+1) = Sum_{k=0..floor(n/2)} binomial(2k, k) = Sum{k=0..n} binomial(k, k/2)*(1+(-1)^k)/2.
%F A100066 a(2n-1) = a(2n) = A006134(n-1) = Sum_{k=0..n}( (2*k)!/(k!)^2 ) for n > 0. - _Alexander Adamchuk_, Feb 23 2007
%F A100066 From _Paul Barry_, Mar 29 2010: (Start)
%F A100066 G.f.: x*(1+x)/((1-x^2)*sqrt(1-4*x^2)) = x/((1-x)*sqrt(1-4*x^2)).
%F A100066 E.g.f.: Integral_{t=0..x} exp(x-t)*Bessel_I(0,2t). (End)
%F A100066 D-finite with recurrence: (n-1)*a(n) - (n-1)*a(n-1) - 4*(n-2)*a(n-2) + 4*(n-2)*a(n-3) = 0. - _R. J. Mathar_, Nov 24 2012
%F A100066 a(n) ~ (3-(-1)^n) * 2^(n+1/2) / (6*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 12 2014
%p A100066 a:=n->sum(binomial(2*j,j), j=0..n): seq(a(n/2), n=-1..40); # _Zerinvary Lajos_, Apr 30 2007
%t A100066 CoefficientList[Series[x/((1-x)*Sqrt[1-4*x^2]), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)
%o A100066 (PARI) my(x='x+O('x^41)); concat(0, Vec(x/((1-x)*sqrt(1-4*x^2)))) \\ _G. C. Greubel_, Jan 30 2017; Mar 17 2025
%o A100066 (Magma)
%o A100066 R<x>:=PowerSeriesRing(Rationals(), 40);
%o A100066 [0] cat Coefficients(R!( x/((1-x)*Sqrt(1-4*x^2)) )); // _G. C. Greubel_, Mar 17 2025
%o A100066 (SageMath)
%o A100066 def b(n): return sum(binomial(2*k,k) for k in range(n+1))
%o A100066 def A100066(n): return b((n-1)//2)
%o A100066 print([A100066(n) for n in range(41)]) # _G. C. Greubel_, Mar 17 2025
%Y A100066 Cf. A000108, A006134, A120582.
%K A100066 nonn,easy
%O A100066 0,4
%A A100066 _Paul Barry_, Nov 02 2004