A235347 Series reversion of x*(1-3*x^2)/(1-x^2) in odd-order powers.

1, 2, 14, 130, 1382, 15906, 192894, 2427522, 31405430, 415086658, 5580629870, 76080887042, 1049295082630, 14613980359010, 205246677882078, 2903566870820610, 41337029956899222, 591796707042765954, 8514525059135909070, 123048063153362454402

Offset

0,2

Comments

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.

Links

Georg Fischer, Table of n, a(n) for n = 0..1000 (first 940 terms from Fung Lam)

Fung Lam, On the Well-posedness of Magnetohydrodynamics Equations for Incompressible Electrically-Conducting Fluids, arXiv:1401.2029 [physics.flu-dyn], 2014-2023.

Formula

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

From Seiichi Manyama, Aug 09 2023: (Start)

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)

Maple

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

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A027307, A107841, A235352 (same except for signs).

Sequence in context: A363983 A258389 A168658 * A235352 A146971 A246481

Adjacent sequences: A235344 A235345 A235346 * A235348 A235349 A235350

Keyword

nonn

Author

Fung Lam, Jan 10 2014

Status

approved