A126932 - OEIS

%I #22 Sep 08 2022 08:45:29

%S 1,4,15,55,199,714,2547,9048,32043,113212,399265,1406079,4946137,

%T 17383162,61048359,214270215,751691811,2636004228,9240836733,

%U 32386215981,113478349989,397544907486,1392493797765,4876916883090,17078574481941,59802541979964

%N Binomial transform of A127358.

%C Hankel transform is (-1)^n.

%C Row sums of the Riordan array ((1-2*x)/(1+x+x^2), x/(1+x+x^2))^(-1). - _Paul Barry_, Nov 06 2008

%H Vincenzo Librandi, <a href="/A126932/b126932.txt">Table of n, a(n) for n = 0..200</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 a(n+1) = 3*a(n) + A059738(n) with a(0)=1.

%F G.f: (sqrt(1-2*x-3*x^2) + 3*(1-3*x))/(2*(2-13*x+21*x^2)). - _Paul Barry_, Nov 06 2008

%F Conjecture: +2*n*a(n) -11*n*a(n-1) +4*(2*n+3)*a(n-2) +21*(n-2)*a(n-3)=0. - _R. J. Mathar_, Nov 24 2012

%F a(n) ~ 3 * 7^n / 2^(n+1). - _Vaclav Kotesovec_, Feb 12 2014

%p seq(coeff(series( (sqrt(1-2*x-3*x^2) + 3*(1-3*x))/(2*(2-13*x+21*x^2)), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Jan 29 2020

%t CoefficientList[Series[(Sqrt[-3*x^2-2*x+1]-3*(3*x-1))/(2*(21*x^2-13*x+2)), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)

%o (PARI) my(x='x+O('x^30)); Vec( (sqrt(1-2*x-3*x^2) + 3*(1-3*x))/(2*(2-13*x+21*x^2)) ) \\ _G. C. Greubel_, Jan 29 2020

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (Sqrt(1-2*x-3*x^2) + 3*(1-3*x))/(2*(2-13*x+21*x^2)) )); // _G. C. Greubel_, Jan 29 2020

%o (Sage)

%o def A126932_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (sqrt(1-2*x-3*x^2) + 3*(1-3*x))/(2*(2-13*x+21*x^2)) ).list()

%o A126932_list(30) # _G. C. Greubel_, Jan 29 2020

%K nonn

%O 0,2

%A _Philippe Deléham_, Mar 17 2007

%E Corrected and extended by _Vincenzo Librandi_, Feb 13 2014