A026854 - OEIS

%I #17 Sep 08 2022 08:44:49

%S 1,3,10,36,136,530,2109,8515,34739,142817,590537,2452639,10221505,

%T 42714623,178888442,750500716,3153137436,13263180550,55844218906,

%U 235323138044,992316962382,4186870456952,17674378119680,74641954142026

%N a(n) = T(2n+1,n+1), T given by A026736.

%H G. C. Greubel, <a href="/A026854/b026854.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: C(x)^2/(1 - x/sqrt(1-4*x)) where C(x) = g.f. for Catalan numbers A000108. - _David Callan_, Jan 16 2016

%F a(n) ~ (3 - sqrt(5))^2 * (2 + sqrt(5))^(n+1) / (4*sqrt(5)). - _Vaclav Kotesovec_, Jul 18 2019

%t CoefficientList[Series[(1-Sqrt[1-4x])^2/(4x^2(1-x/Sqrt[1-4x])), {x, 0, 30}], x] (* _David Callan_, Jan 16 2016 *)

%o (PARI) my(x='x+O('x^30)); Vec( (1-sqrt(1-4*x))^2/(4*x^2*(1-x/sqrt(1-4*x))) ) \\ _G. C. Greubel_, Jul 21 2019

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-Sqrt(1-4*x))^2/(4*x^2*(1-x/Sqrt(1-4*x))) )); // _G. C. Greubel_, Jul 21 2019

%o (Sage) ((1-sqrt(1-4*x))^2/(4*x^2*(1-x/sqrt(1-4*x)))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 21 2019

%Y Cf. A000108, A026736.

%K nonn

%O 0,2

%A _Clark Kimberling_