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

%I #18 Nov 09 2024 07:01:19

%S -1,2,3,10,43,210,1106,6120,35059,206050,1235130,7520860,46385822,

%T 289160620,1818956184,11531388580,73599630147,472547337650,

%U 3049962382850,19777692906300,128789274380746,841837751136060,5521644155232972,36330127521846080,239722100410099118

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

%H G. C. Greubel, <a href="/A243642/b243642.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (Sum_{k=0..(n-1)} binomial(3*n-3,n-4*k)*binomial(n+k-2,k))/(n-1), n>1, a(0) = -1, a(1)=2.

%F A(x) = -1/C(x)*S(x*C(x)), where C(x) is g.f. A000108, S(x) is g.f. A001003.

%F a(n) ~ sqrt((27-19*sqrt(2))/7) * (7+5*sqrt(2))^n / (sqrt(Pi) * n^(3/2) * 2^(n-3/4)). - _Vaclav Kotesovec_, Jun 15 2014

%F Conjecture D-finite with recurrence: +16*n*(n-1)*(n-2)*a(n) -4*(n-1)*(n-2)*(43*n-54)*a(n-1) +2*(n-2)*(52*n^2+695*n-2439)*a(n-2) +(3479*n^3-42714*n^2+169789*n-219234)*a(n-3) +2*(-4633*n^3+64020*n^2-293825*n+447594)*a(n-4) +4*(2*n-11)*(296*n^2-3643*n+11187)*a(n-5) +24*(n-7)*(2*n-11)*(2*n-13)*a(n-6)=0. - _R. J. Mathar_, Jan 25 2020

%t CoefficientList[Series[8*x/(-3+Sqrt[1-4*x] + Sqrt[-6+10*Sqrt[1-4*x] - 4*x]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 15 2014 *)

%o (Maxima)

%o a(n):=if n=0 then -1 else if n=1 then 2 else sum(binomial(3*n-3,n-4*k)*binomial(n+k-2,k),k,0,n-1)/(n-1);

%o (PARI) my(x='x+O('x^50)); Vec(-(4*x)/((1-sqrt(1-4*x))/2-sqrt(-3*(1-sqrt(1-4*x))+(1-sqrt(1-4*x))^2/4+1)+1)) \\ _G. C. Greubel_, Jun 02 2017

%Y Cf. A000108, A001003.

%K sign,changed

%O 0,2

%A _Vladimir Kruchinin_, Jun 08 2014