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%I #25 Mar 24 2023 14:59:47
%S 1,2,8,42,248,1570,10416,71474,503088,3612226,26353720,194806458,
%T 1455874792,10982013250,83504148192,639360351074,4925190101600,
%U 38144591091970,296837838901992,2319880586624714,18200693844341720,143294043656426082,1131747417739664528
%N Series reversion of x*(1-2*x-x^2)/(1-x^2).
%C Derived series from A107841. The reversion has a quadratic power in x in the denominator. The general form reads x*(1-p*x-q*x^2)/(1-q*x^2).
%H Fung Lam, <a href="/A235350/b235350.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: (exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v - 2/3)/x, where i=sqrt(-1),
%F u = 1/3*(-17+3*x-6*x^2+x^3+3*sqrt(-6+54*x-30*x^2+18*x^3-3*x^4))^(1/3), and
%F v = 1/3*(-17+3*x-6*x^2+x^3-3*sqrt(-6+54*x-30*x^2+18*x^3-3*x^4))^(1/3).
%F First few terms can be obtained by Maclaurin's expansion of G.f.
%F D-finite with recurrence 6*n*(n-1)*a(n) -(n-1)*(52*n-75)*a(n-1) +(2*n+3)*(5*n-11)*a(n-2) +2*(5*n^2-62*n+150)*a(n-3) +(-13*n^2+130*n-321)*a(n-4) +(7*n-37)*(n-6)*a(n-5) -(n-6)*(n-7)*a(n-6)=0. - _R. J. Mathar_, Mar 24 2023
%t Rest[CoefficientList[InverseSeries[Series[x*(1-2*x-x^2)/(1-x^2), {x, 0, 20}], x],x]] (* _Vaclav Kotesovec_, Jan 29 2014 *)
%o (Python)
%o # a235350. The list a has been calculated (len(a)>=3).
%o m = len(a)
%o d = 0
%o for i in range (1,m+3):
%o ....for j in range (1,m+3):
%o ........if (i+j)%m ==0 and (i+j) <= m:
%o ............d = d + a[i-1]*a[j-1]
%o f = 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 ............f = f + a[i-1]*a[j-1]
%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,ip):
%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 = g + 2*f - d
%o # a235350.
%o (PARI) Vec(serreverse(x*(1-2*x-x^2)/(1-x^2)+O(x^66))) \\ _Joerg Arndt_, Jan 17 2014
%Y Cf. A107841.
%K nonn,easy
%O 1,2
%A _Fung Lam_, Jan 16 2014
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