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A235350 Series reversion of x*(1-2*x-x^2)/(1-x^2).

%I A235350 #33 Sep 04 2024 15:25:51
%S A235350 1,2,8,42,248,1570,10416,71474,503088,3612226,26353720,194806458,
%T A235350 1455874792,10982013250,83504148192,639360351074,4925190101600,
%U A235350 38144591091970,296837838901992,2319880586624714,18200693844341720,143294043656426082,1131747417739664528
%N A235350 Series reversion of x*(1-2*x-x^2)/(1-x^2).
%C A235350 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 A235350 Fung Lam, <a href="/A235350/b235350.txt">Table of n, a(n) for n = 1..1000</a>
%F A235350 G.f.: (exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v - 2/3)/x, where i=sqrt(-1),
%F A235350 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 A235350 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 A235350 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 A235350 Rest[CoefficientList[InverseSeries[Series[x*(1-2*x-x^2)/(1-x^2), {x, 0, 20}], x],x]] (* _Vaclav Kotesovec_, Jan 29 2014 *)
%o A235350 (Python)
%o A235350 a = [0, 1]
%o A235350 for n in range(20):
%o A235350     m = len(a)
%o A235350     d = 0
%o A235350     for i in range (1, m):
%o A235350         for j in range (1, m):
%o A235350             if (i+j)%(m-1) == 0 and (i+j) < m:
%o A235350                 d += a[i]*a[j]
%o A235350     f = 0
%o A235350     for i in range (1, m):
%o A235350         for j in range (1, m):
%o A235350             if (i+j)%m == 0 and (i+j) <= m:
%o A235350                 f += a[i]*a[j]
%o A235350     g = 0
%o A235350     for i in range (1, m):
%o A235350         for j in range (1, m):
%o A235350             for k in range (1, m):
%o A235350                 if (i+j+k)%m == 0 and (i+j+k) <= m:
%o A235350                     g += a[i]*a[j]*a[k]
%o A235350     y = g + 2*f - d
%o A235350     a.append(y)
%o A235350 print(a[1:]) # Edited by _Andrey Zabolotskiy_, Sep 04 2024
%o A235350 (PARI) Vec(serreverse(x*(1-2*x-x^2)/(1-x^2)+O(x^66))) \\ _Joerg Arndt_, Jan 17 2014
%Y A235350 Cf. A107841, A235347, A235348, A235349, A235351, A235352.
%K A235350 nonn,easy
%O A235350 1,2
%A A235350 _Fung Lam_, Jan 16 2014