%I #38 Jan 30 2020 21:29:17
%S 0,0,1,4,15,57,217,828,3169,12165,46827,180701,698867,2708307,
%T 10514331,40885356,159216543,620845293,2423825649,9473195889,
%U 37061983617,145131715707,568808493081,2231063305461,8757391892965,34397931629763,135196161588037,531682892209431
%N Expansion of (1 - 2*x - sqrt(1-4*x))/(4*x^2 + sqrt(1-4*x)*(3*x+1) - 5*x + 1).
%H G. C. Greubel, <a href="/A242781/b242781.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=1..ceiling((n+1)/3)} binomial(2*n-3*k+1,n-3*k+1)), n>0, a(0)=0.
%F G.f.: log'(1/(1-x^3*C(x)^3))/3, where C(x) is g.f. of A000108.
%F a(n) ~ 2^(2*n+1)/(7*sqrt(Pi*n)). - _Vaclav Kotesovec_, May 24 2014
%F Conjecture D-finite with recurrence: 3*(n+1)*a(n) -18*n*a(n-1) +2*(11*n-13)*a(n-2) +6*(n-7)*a(n-3) +(7*n-9)*a(n-4) +2*(2*n-7)*a(n-5)=0. - _R. J. Mathar_, Jan 25 2020
%t CoefficientList[Series[(1-2*x-Sqrt[1-4*x])/(4*x^2+Sqrt[1-4*x]*(3*x+1)-5*x+1), {x, 0, 20}], x] (* _Vaclav Kotesovec_, May 24 2014 *)
%o (Maxima)
%o a(n):=sum(binomial(2*n-3*k+1,n-3*k+1),k,1,ceiling((n+1)/3));
%o (PARI) x='x+O('x^50); concat([0,0], Vec((1-2*x-sqrt(1-4*x))/(4*x^2 + sqrt(1-4*x)*(3*x+1)-5*x+1))) \\ _G. C. Greubel_, Jun 01 2017
%Y Cf. A000108.
%K nonn
%O 0,4
%A _Vladimir Kruchinin_, May 24 2014
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