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A235349
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Series reversion of x*(1-x-2*x^2)/(1-x).
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2
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0, 1, 0, 2, 2, 14, 30, 146, 434, 1862, 6470, 26586, 99946, 406366, 1593774, 6492450, 26100578, 106979894, 436906902, 1803472874, 7446478746, 30945624910, 128821054846, 538584390834, 2256485249682, 9483898177574
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OFFSET
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0,4
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COMMENTS
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Derived turbulence series from A235347.
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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 - 1/6 )/x, where i=sqrt(-1),
u = 1/6*(-10-63*x+3*sqrt(-24*x^3+357*x^2+42*x-27))^(1/3), and
v = 1/6*(-10-63*x-3*sqrt(-24*x^3+357*x^2+42*x-27))^(1/3).
First few terms can be obtained by Maclaurin's expansion of G.f.
a(n) ~ sqrt((1-s)^3 / (2*s*(3 - 3*s + s^2))) / (2*sqrt(Pi) * n^(3/2) * r^(n-1/2)), where s = 0.31472177038151893868... is the root of the equation 1-2*s-5*s^2+4*s^3 = 0, and r = s*(1-s-2*s^2)/(1-s) = 0.22374229727550306625... - Vaclav Kotesovec, Jan 23 2014
D-finite with recurrence 117*n*(n-1)*a(n) -7*(n-1)*(35*n-66)*a(n-1) +21*(-69*n^2+269*n-254)*a(n-2) +(937*n^2-6403*n+10920)*a(n-3) -28*(n-4)*(2*n-9)*a(n-4)=0. - R. J. Mathar, Mar 24 2023
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MATHEMATICA
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CoefficientList[InverseSeries[Series[x*(1-x-2*x^2)/(1-x), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Jan 22 2014 *)
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PROG
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(Python)
# a235349. The list a has been calculated (len(a)>=3).
m = len(a)
d = 0
for i in range (1, m+1):
....for j in range (1, m+1):
........if (i+j)%(m+1) ==0 and (i+j) <= (m+1):
............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 = 2*g + d - a[m-1]
# a235349.
(PARI) Vec(serreverse(x*(1-x-2*x^2)/(1-x)+O(x^66))) \\ Joerg Arndt, Jan 17 2014
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Prepended a(0)=0 to adapt to offset 0, Joerg Arndt, Jan 23 2014
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STATUS
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approved
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