|
%I #34 Feb 14 2022 21:24:17
%S 1,3,8,21,54,138,350,885,2230,5610,14088,35346,88596,221952,555738,
%T 1391061,3480870,8708610,21783680,54483510,136254964,340729788,
%U 852000828,2130354786,5326563004
%N a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*2^(n-k).
%C 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)/(1-2x) 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 Harvey P. Dale, <a href="/A127358/b127358.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 G.f.: (1/sqrt(1 - 4x^2))(1 + x*c(x^2))/(1 - 2*x*c(x^2)).
%F a(n) = 2*a(n-1) + A054341(n-1). a(n) = Sum_{k=0..n} A126075(n,k). - _Philippe Deléham_, Mar 03 2007
%F a(n) = Sum_{k=0..n} A061554(n,k)*2^k. - _Philippe Deléham_, Dec 04 2009
%F From _Gary W. Adamson_, Sep 07 2011: (Start)
%F a(n) is the sum of top row terms of M^n, M is an infinite square production matrix as follows:
%F 2, 1, 0, 0, 0, ...
%F 1, 0, 1, 0, 0, ...
%F 0, 1, 0, 1, 0, ...
%F 0, 0, 1, 0, 1, ...
%F 0, 0, 0, 1, 0, ...
%F ... (End)
%F Conjecture: 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 a(n) ~ 3 * 5^n / 2^(n+1). - _Vaclav Kotesovec_, Feb 13 2014
%e a(3) = 21 = (12 + 6 + 2 + 1), where the top row of M^3 = (12, 6, 2, 1).
%t Table[Sum[Binomial[n,Floor[k/2]]2^(n-k),{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Jun 03 2012 *)
%t 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 (PARI) 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 Cf. A107430. - _Philippe Deléham_, Sep 16 2009
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jan 11 2007
|
