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A235349 Series reversion of x*(1-x-2*x^2)/(1-x).

%I A235349 #27 Sep 04 2024 16:43:44
%S A235349 0,1,0,2,2,14,30,146,434,1862,6470,26586,99946,406366,1593774,6492450,
%T A235349 26100578,106979894,436906902,1803472874,7446478746,30945624910,
%U A235349 128821054846,538584390834,2256485249682,9483898177574
%N A235349 Series reversion of x*(1-x-2*x^2)/(1-x).
%C A235349 Derived turbulence series from A235347.
%H A235349 Fung Lam, <a href="/A235349/b235349.txt">Table of n, a(n) for n = 0..1000</a>
%F A235349 G.f.: ( exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v - 1/6 )/x, where i=sqrt(-1),
%F A235349 u = 1/6*(-10-63*x+3*sqrt(-24*x^3+357*x^2+42*x-27))^(1/3), and
%F A235349 v = 1/6*(-10-63*x-3*sqrt(-24*x^3+357*x^2+42*x-27))^(1/3).
%F A235349 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 A235349 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 A235349 CoefficientList[InverseSeries[Series[x*(1-x-2*x^2)/(1-x), {x, 0, 20}], x],x] (* _Vaclav Kotesovec_, Jan 22 2014 *)
%o A235349 (Python)
%o A235349 a = [0, 1]
%o A235349 for n in range(20):
%o A235349     m = len(a)
%o A235349     d = 0
%o A235349     for i in range (1, m):
%o A235349         for j in range (1, m):
%o A235349             if (i+j)%m == 0 and (i+j) <= m:
%o A235349                 d += a[i]*a[j]
%o A235349     g = 0
%o A235349     for i in range (1, m-1):
%o A235349         for j in range (1, m-1):
%o A235349             for k in range (1, m-1):
%o A235349                 if (i+j+k)%m == 0 and (i+j+k) <= m:
%o A235349                     g += a[i]*a[j]*a[k]
%o A235349     y = 2*g + d - a[m-1]
%o A235349     a.append(y)
%o A235349 print(a)
%o A235349 (PARI) Vec(serreverse(x*(1-x-2*x^2)/(1-x)+O(x^66))) \\ _Joerg Arndt_, Jan 17 2014
%Y A235349 Cf. A235347, A235348, A235350, A235351, A235352.
%K A235349 nonn,easy
%O A235349 0,4
%A A235349 _Fung Lam_, Jan 16 2014
%E A235349 Prepended a(0)=0 to adapt to offset 0, _Joerg Arndt_, Jan 23 2014
%E A235349 b-file shifted for offset 0, _Vaclav Kotesovec_, Jan 23 2014