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A235347 Series reversion of x*(1-3*x^2)/(1-x^2) in odd-order powers. 4
1, 2, 14, 130, 1382, 15906, 192894, 2427522, 31405430, 415086658, 5580629870, 76080887042, 1049295082630, 14613980359010, 205246677882078, 2903566870820610, 41337029956899222, 591796707042765954, 8514525059135909070, 123048063153362454402 (list; graph; refs; listen; history; text; internal format)
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

Fung Lam, Table of n, a(n) for n = 0..1000

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

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.

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

(Python)

# a235347. The list a has been calculated (len(a)>=3)

m = len(a)+1

d = 0

for i in range (1, m):

....for j in range (1, m):

........if (i+j)%m ==0 and (i+j) <= m:

............d = d + a[i-1]*a[j-1]

g = 0

for i in range (1, m):

....for j in range (1, m):

........for k in range (1, m):

............if (i+j+k)%(m+1) ==0 and (i+j+k) <= (m+1):

................g = g + a[i-1]*a[j-1]*a[k-1]

y = 3*g - d

# a235347.

(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. A235352 (same except for signs).

Sequence in context: A331397 A258389 A168658 * A235352 A146971 A246481

Adjacent sequences:  A235344 A235345 A235346 * A235348 A235349 A235350

KEYWORD

nonn

AUTHOR

Fung Lam, Jan 10 2014

STATUS

approved

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Last modified May 20 23:05 EDT 2022. Contains 353886 sequences. (Running on oeis4.)