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 A127363 #14 Jul 31 2025 16:52:04
%S A127363 1,-3,14,-57,246,-1038,4424,-18777,79846,-339258,1442004,-6128202,
%T A127363 26045436,-110691948,470442924,-1999378137,8497365126,-36113785698,
%U A127363 153483619604,-652305322542,2772297736276,-11782265148228,50074627320864,-212817165231882,904472953925596
%N A127363 a(n) = Sum_{k=0..n} C(n,floor(k/2))*(-4)^(n-k).
%C A127363 Hankel transform is 5^n. In general, for r>=0, the sequence given by Sum_{k=0..n} C(n,floor(k/2))*(-r)^(n-k) has Hankel transform (r+1)^n. The sequence is the image of the sequence with g.f. (1+x)/(1+4x) under the Chebyshev mapping g(x)->(1/sqrt(1-4x^2))g(xc(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.
%H A127363 Vincenzo Librandi, <a href="/A127363/b127363.txt">Table of n, a(n) for n = 0..200</a>
%F A127363 G.f.: (1/sqrt(1-4x^2))(1+x*c(x^2))/(1+4*x*c(x^2)).
%F A127363 Conjecture: 4*n*a(n) +(17*n-8)*a(n-1) +2*(-8*n-1)*a(n-2) +68*(-n+2)*a(n-3)=0. - _R. J. Mathar_, Nov 24 2012
%F A127363 a(n) ~ (-1)^n * 3 * 17^n / 4^(n+1). - _Vaclav Kotesovec_, Feb 12 2014
%t A127363 CoefficientList[Series[1/Sqrt[1-4*x^2] * (1+x*(1-Sqrt[1-4*x^2]) / (2*x^2)) / (1+4*x*(1-Sqrt[1-4*x^2])/(2*x^2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)
%t A127363 Table[Sum[Binomial[n,Floor[k/2]](-4)^(n-k),{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Jul 31 2025 *)
%Y A127363 Cf. A000108.
%K A127363 easy,sign
%O A127363 0,2
%A A127363 _Paul Barry_, Jan 11 2007
%E A127363 More terms from _Vincenzo Librandi_, Feb 13 2014