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%I #42 Aug 09 2023 11:38:54
%S 1,2,14,130,1382,15906,192894,2427522,31405430,415086658,5580629870,
%T 76080887042,1049295082630,14613980359010,205246677882078,
%U 2903566870820610,41337029956899222,591796707042765954,8514525059135909070,123048063153362454402
%N Series reversion of x*(1-3*x^2)/(1-x^2) in odd-order powers.
%C This sequence is implied in the solutions of magnetohydrodynamics equations in R^3 for incompressible, electrically-conducting fluids in the presence of a strong Lorentz force. a(n) = numbers of allowable magneto-vortical eddies in terms of initial conditions.
%H Georg Fischer, <a href="/A235347/b235347.txt">Table of n, a(n) for n = 0..1000</a> (first 940 terms from _Fung Lam_)
%H Fung Lam, <a href="http://arxiv.org/abs/1401.2029">On the Well-posedness of Magnetohydrodynamics Equations for Incompressible Electrically-Conducting Fluids</a>, arXiv:1401.2029 [physics.flu-dyn], 2014-2023.
%F G.f.: (exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v + x/9)/x, where i=sqrt(-1),
%F u = 1/9*(x^3 - 108 *x + 9*sqrt(-9 + 141*x^2 - 3*x^4))^(1/3), and
%F v = 1/9*(x^3 - 108 *x - 9*sqrt(-9 + 141*x^2 - 3*x^4))^(1/3).
%F First few a(n)'s can be obtained by either considering Maclaurin's expansion of G.f. or evaluating the coefficient of x^(n) in 2*sum{j,1,n} ((sum{k,1,n} a(k) x^(2*k-1))^(2*j+1)), a(1)=1, with offset by 1.
%F D-finite with recurrence 12*n*(2*n+1)*a(n) +(-382*n^2+391*n-90)*a(n-1) +3*(34*n^2-132*n+125)*a(n-2) -(2*n-5)*(n-3)*a(n-3)=0. - _R. J. Mathar_, Mar 24 2023
%F From _Seiichi Manyama_, Aug 09 2023: (Start)
%F a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(2*n+k+1,n) / (2*n+k+1).
%F a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-k) for n > 0.
%F a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(2*n,k-1) for n > 0. (End)
%p Order := 60 ;
%p solve(series(x*(1-3*x^2)/(1-x^2),x)=y,x) ;
%p convert(%,polynom) ;
%p seq(coeff(%,y,2*i+1),i=0..Order/2) ; # _R. J. Mathar_, Jul 20 2023
%t Table[(CoefficientList[InverseSeries[Series[x*(1-3*x^2)/(1-x^2),{x,0,40}],x],x])[[n]],{n,2,40,2}] (* _Vaclav Kotesovec_, Jan 29 2014 *)
%o (PARI) v=Vec( serreverse(x*(1-3*x^2)/(1-x^2) +O(x^66) ) ); vector(#v\2,j,v[2*j-1]) \\ _Joerg Arndt_, Jan 14 2014
%Y Cf. A027307, A107841, A235352 (same except for signs).
%K nonn
%O 0,2
%A _Fung Lam_, Jan 10 2014
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