A234269 - OEIS

%I #20 Sep 08 2022 08:46:06

%S 0,0,1,1,4,7,18,39,90,206,470,1085,2492,5762,13314,30849,71556,166210,

%T 386562,899976,2097524,4892966,11423984,26693381,62417940,146053272,

%U 341970538,801168316,1878016792,4404544926,10335098184,24262063281,56980852484,133877548896

%N Expansion of (1-2*x^2-sqrt(1-4*x^2-4*x^3))/(2*x*sqrt(1-4*x^2-4*x^3)).

%H Alois P. Heinz, <a href="/A234269/b234269.txt">Table of n, a(n) for n = 0..1000</a>

%H J.-L. Baril, J.-M. Pallo, <a href="http://jl.baril.u-bourgogne.fr/Motzkin.pdf">Motzkin subposet and Motzkin geodesics in Tamari lattices</a>, 2013.

%F Conjecture D-finite with recurrence: -(n+1)*(2*n-5)*a(n) -2*n*(n-4)*a(n-1) +4*(n-1)*(2*n-3)*a(n-2) +2*(8*n^2-32*n+29)*a(n-3) +4*(n-2)*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Jan 24 2020

%t CoefficientList[Series[(1 - 2 x^2 - Sqrt[1 - 4 x^2 - 4 x^3]) / (2 x Sqrt[1 - 4 x^2 - 4 x^3]), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jan 25 2020 *)

%o (Magma)  m:=50; R<x>:=PowerSeriesRing(Rationals(), m); [0,0] cat  Coefficients(R!((1 - 2*x^2 - Sqrt(1 - 4*x^2 - 4*x^3)) / (2*x*Sqrt(1 - 4*x^2 - 4*x^3)))); // _Vincenzo Librandi_, Jan 25 2020

%o (PARI) seq(n)={Vec((1-2*x^2-sqrt(1-4*x^2-4*x^3 + O(x^2*x^n)))/(2*x*sqrt(1-4*x^2-4*x^3 + O(x*x^n))), -(n+1))} \\ _Andrew Howroyd_, Jan 25 2020

%Y Cf. A234270.

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Dec 24 2013

%E Offset changed to 0 and more terms from _Alois P. Heinz_, Nov 16 2015