# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a127358 Showing 1-1 of 1 %I A127358 #34 Feb 14 2022 21:24:17 %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 %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)/(1-2x) 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 A127358 Harvey P. Dale, Table of n, a(n) for n = 0..1000 %H A127358 Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, 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 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 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) 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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE