A235349 - OEIS

%I #27 Sep 04 2024 16:43:44

%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 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 a = [0, 1]

%o for n in range(20):

%o     m = len(a)

%o     d = 0

%o     for i in range (1, m):

%o         for j in range (1, m):

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

%o                 d += a[i]*a[j]

%o     g = 0

%o     for i in range (1, m-1):

%o         for j in range (1, m-1):

%o             for k in range (1, m-1):

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

%o                     g += a[i]*a[j]*a[k]

%o     y = 2*g + d - a[m-1]

%o     a.append(y)

%o print(a)

%o (PARI) Vec(serreverse(x*(1-x-2*x^2)/(1-x)+O(x^66))) \\ _Joerg Arndt_, Jan 17 2014

%Y Cf. A235347, A235348, A235350, A235351, A235352.

%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