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A196868 G.f. A(x) satisfies: A(x)^2 + A(-x)^2 = 2 and A(x)^3 - A(-x)^3 = 36*x.

%I A196868 #7 Apr 04 2022 19:50:28
%S A196868 1,6,-18,144,-1026,10368,-91044,995328,-9630090,109486080,-1120744188,
%T A196868 13042778112,-138540597588,1637370298368,-17853248637000,
%U A196868 213325958873088,-2371846639850586,28573129903177728,-322526246042905740,3910007249908531200,-44670671340291807228
%N A196868 G.f. A(x) satisfies: A(x)^2 + A(-x)^2 = 2 and A(x)^3 - A(-x)^3 = 36*x.
%F A196868 From _Paul D. Hanna_, Apr 04 2022: (Start)
%F A196868 a(2*n+1) = 6 * 24^n * binomial(3*n+1,n)/(3*n+1) for n >= 0.
%F A196868 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A196868 (1) (A(x)^2 + A(-x)^2)/2 = 1.
%F A196868 (2) (A(x)^3 - A(-x)^3)/2 = 18*x.
%F A196868 (3) (A(x) - A(-x))/2 = 6*Series_Reversion(x - 24*x^3).
%F A196868 (4) (A(x) + A(-x))/2 = sqrt( (1 + A(x)*A(-x))/2 ).
%F A196868 (5) A(x)*A(-x) = 1 - 72*Series_Reversion(x - 24*x^3)^2.
%F A196868 (6) A(x) = B(x) + sqrt(1 - B(x)^2), where B(x) = 6*Series_Reversion(x - 24*x^3).
%F A196868 (End)
%e A196868 G.f.: A(x) = 1 + 6*x - 18*x^2 + 144*x^3 - 1026*x^4 + 10368*x^5 +...
%e A196868 where
%e A196868 A(x)^2 = 1 + 12*x + 72*x^3 + 3240*x^5 + 229392*x^7 + 20083464*x^9 +...
%e A196868 A(x)^3 = 1 + 18*x + 54*x^2 + 1134*x^4 + 63180*x^6 + 4903254*x^8 +...
%o A196868 (PARI) {a(n)=local(A=[1,6]);for(k=2,n,A=concat(A,0);if(k%2==0,A[#A]=-Vec(Ser(A)^2)[#A]/2,A[#A]=-Vec(Ser(A)^3)[#A]/3));A[n+1]}
%o A196868 (PARI) /* Using series reversion: */
%o A196868 {a(n) = my(A, B = 6*serreverse(x - 24*x^3 +x^2*O(x^n)) );
%o A196868 A = B + sqrt(1 - B^2); polcoeff(A,n)}
%o A196868 for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Apr 04 2022
%Y A196868 Cf. A196869, A193618, A193619, A196864, A196865, A196866, A196867.
%K A196868 sign
%O A196868 0,2
%A A196868 _Paul D. Hanna_, Oct 06 2011