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 A127358 #39 Jan 26 2025 20:27:03
%S A127358 1,3,8,21,54,138,350,885,2230,5610,14088,35346,88596,221952,555738,
%T A127358 1391061,3480870,8708610,21783680,54483510,136254964,340729788,
%U A127358 852000828,2130354786,5326563004,13317759588,33296999120,83247698100,208129274400,520343244300
%N A127358 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*2^(n-k).
%C A127358 Hankel transform is (-1)^n. In general, given r >= 0, the sequence given by Sum_{k=0..n} binomial(n, floor(k/2))*r^(n-k) has Hankel transform (1-r)^n. The sequence is the image of the sequence with g.f. (1+x)/(1-2*x) under the Chebyshev mapping g(x) -> (1/sqrt(1-4*x^2))*g(x*c(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.
%H A127358 Harvey P. Dale, <a href="/A127358/b127358.txt">Table of n, a(n) for n = 0..1000</a>
%H A127358 Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%F A127358 G.f.: (1/sqrt(1 - 4x^2))(1 + x*c(x^2))/(1 - 2*x*c(x^2)).
%F A127358 a(n) = 2*a(n-1) + A054341(n-1). a(n) = Sum_{k=0..n} A126075(n,k). - _Philippe Deléham_, Mar 03 2007
%F A127358 a(n) = Sum_{k=0..n} A061554(n,k)*2^k. - _Philippe Deléham_, Dec 04 2009
%F A127358 From _Gary W. Adamson_, Sep 07 2011: (Start)
%F A127358 a(n) is the sum of top row terms of M^n, M is an infinite square production matrix as follows:
%F A127358 2, 1, 0, 0, 0, ...
%F A127358 1, 0, 1, 0, 0, ...
%F A127358 0, 1, 0, 1, 0, ...
%F A127358 0, 0, 1, 0, 1, ...
%F A127358 0, 0, 0, 1, 0, ...
%F A127358 ... (End)
%F A127358 D-finite with recurrence 2*n*a(n) + (-5*n-4)*a(n-1) + 2*(-4*n+13)*a(n-2) + 20*(n-2)*a(n-3) = 0. - _R. J. Mathar_, Nov 30 2012
%F A127358 a(n) ~ 3 * 5^n / 2^(n+1). - _Vaclav Kotesovec_, Feb 13 2014
%e A127358 a(3) = 21 = (12 + 6 + 2 + 1), where the top row of M^3 = (12, 6, 2, 1).
%t A127358 Table[Sum[Binomial[n,Floor[k/2]]2^(n-k),{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Jun 03 2012 *)
%t A127358 CoefficientList[Series[(1 + 2*x - Sqrt[1 - 4*x^2])/(2*Sqrt[1 - 4*x^2]*(x - 1 + Sqrt[1 - 4*x^2])), {x, 0, 50}], x] (* _G. C. Greubel_, May 22 2017 *)
%o A127358 (PARI) my(x='x+O('x^50)); Vec((1 + 2*x - sqrt(1 - 4*x^2))/(2*sqrt(1 - 4*x^2)*(x - 1 + sqrt(1 - 4*x^2)))) \\ _G. C. Greubel_, May 22 2017
%Y A127358 Cf. A107430. - _Philippe Deléham_, Sep 16 2009
%K A127358 easy,nonn
%O A127358 0,2
%A A127358 _Paul Barry_, Jan 11 2007