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%I #13 Dec 24 2023 08:47:32
%S 1,2,6,16,52,152,512,1568,5392,16992,59232,190336,669952,2183680,
%T 7742464,25512448,90974464,302368256,1083175424,3625435136,
%U 13036688384,43889186816,158323564544,535639556096,1937483350016,6582584115200,23865932414976,81381420826624,295661476642816
%N Expansion of g.f. A(x) = B(x^2)/(1 - 2*x*B(x^2)) where B(x) = 1 + 2*x*B(x)^3 is the g.f. of A153231.
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F (1) A(x) = B(x^2)/(1 - 2*x*B(x^2)) where B(x) = 1 + 2*x*B(x)^3 is the g.f. of A153231.
%F (2) A(x) = 1/(1-2*x - Sum_{n>=1} 2^n * binomial(3*n-1,n)/(3*n-1) * x^(2*n) ).
%F (3) A(x) = 1/(-2*x + x/Series_Reversion( x - 2*x^3 )).
%F (4) [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n-1)^(n-2) for n > 1.
%F (5) [x^(n-1)] (1 + n*x*A(x))^n / A(x)^n = -n*(n-2)^(n-2) for n > 1.
%F (6) [x^(n-1)] (1 + (n-2)*x*A(x))^n / A(x)^n = -n*(5*n-14)*(n-4)^(n-3) for n >= 1.
%F a(n) ~ (15*sqrt(3/2)/2 + 9 + (15*sqrt(3/2)/2 - 9)*(-1)^n) * 3^(3*n/2) / (sqrt(Pi) * n^(3/2) * 2^(n/2)). - _Vaclav Kotesovec_, Dec 24 2023
%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 16*x^3 + 52*x^4 + 152*x^5 + 512*x^6 + 1568*x^7 + 5392*x^8 + 16992*x^9 + 59232*x^10 + ...
%e where
%e 1/A(x) = 1 - 2*x - 2*x^2 - 8*x^4 - 56*x^6 - 480*x^8 - 4576*x^10 - 46592*x^12 - ... - 2^n*binomial(3*n-1,n)/(3*n-1) * x^(2*n) - ...
%t CoefficientList[1/(-2*x + x/InverseSeries[Series[x - 2*x^3, {x, 0, 30}], x]), x] (* _Vaclav Kotesovec_, Dec 24 2023 *)
%o (PARI) {a(n) = my(A = 1/(-2*x + x/serreverse(x - 2*x^3 + O(x^(n+2))))); polcoeff(A,n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A;
%o A[#A] = polcoeff( (1 + (m+1)*x*Ser(A))^m / Ser(A)^m , m-1)/m - (m-1)^(m-2) ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A153231.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 20 2023
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