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A235347
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Series reversion of x*(1-3*x^2)/(1-x^2) in odd-order powers.
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7
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1, 2, 14, 130, 1382, 15906, 192894, 2427522, 31405430, 415086658, 5580629870, 76080887042, 1049295082630, 14613980359010, 205246677882078, 2903566870820610, 41337029956899222, 591796707042765954, 8514525059135909070, 123048063153362454402
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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G.f.: (exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v + x/9)/x, where i=sqrt(-1),
u = 1/9*(x^3 - 108 *x + 9*sqrt(-9 + 141*x^2 - 3*x^4))^(1/3), and
v = 1/9*(x^3 - 108 *x - 9*sqrt(-9 + 141*x^2 - 3*x^4))^(1/3).
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.
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
a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(2*n+k+1,n) / (2*n+k+1).
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.
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)
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MAPLE
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Order := 60 ;
solve(series(x*(1-3*x^2)/(1-x^2), x)=y, x) ;
convert(%, polynom) ;
seq(coeff(%, y, 2*i+1), i=0..Order/2) ; # R. J. Mathar, Jul 20 2023
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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