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%I #22 Mar 24 2023 14:51:11
%S 0,1,0,2,2,14,30,146,434,1862,6470,26586,99946,406366,1593774,6492450,
%T 26100578,106979894,436906902,1803472874,7446478746,30945624910,
%U 128821054846,538584390834,2256485249682,9483898177574
%N Series reversion of x*(1-x-2*x^2)/(1-x).
%C Derived turbulence series from A235347.
%H Fung Lam, <a href="/A235349/b235349.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: ( exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v - 1/6 )/x, where i=sqrt(-1),
%F u = 1/6*(-10-63*x+3*sqrt(-24*x^3+357*x^2+42*x-27))^(1/3), and
%F v = 1/6*(-10-63*x-3*sqrt(-24*x^3+357*x^2+42*x-27))^(1/3).
%F First few terms can be obtained by Maclaurin's expansion of G.f.
%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
%F 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
%t CoefficientList[InverseSeries[Series[x*(1-x-2*x^2)/(1-x), {x, 0, 20}], x],x] (* _Vaclav Kotesovec_, Jan 22 2014 *)
%o (Python)
%o # a235349. The list a has been calculated (len(a)>=3).
%o m = len(a)
%o d = 0
%o for i in range (1,m+1):
%o ....for j in range (1,m+1):
%o ........if (i+j)%(m+1) ==0 and (i+j) <= (m+1):
%o ............d = d + a[i-1]*a[j-1]
%o g = 0
%o for i in range (1,m):
%o ....for j in range (1,m):
%o ........for k in range (1,m):
%o ............if (i+j+k)%(m+1) ==0 and (i+j+k) <= (m+1):
%o ................g = g + a[i-1]*a[j-1]*a[k-1]
%o y = 2*g + d - a[m-1]
%o # a235349.
%o (PARI) Vec(serreverse(x*(1-x-2*x^2)/(1-x)+O(x^66))) \\ _Joerg Arndt_, Jan 17 2014
%Y Cf. A235347.
%K nonn,easy
%O 0,4
%A _Fung Lam_, Jan 16 2014
%E Prepended a(0)=0 to adapt to offset 0, _Joerg Arndt_, Jan 23 2014
%E b-file shifted for offset 0, _Vaclav Kotesovec_, Jan 23 2014
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