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A352383 G.f. A(x) satisfies: A(x) = (1 + 2*x*A(x)^3)^2 / (1 + x*A(x)^3).

%I A352383 #9 Mar 15 2022 08:20:55
%S A352383 1,3,28,350,5020,78023,1278340,21740636,380161308,6792111260,
%T A352383 123448657904,2275311657814,42427160829508,798933055335618,
%U A352383 15171376583787800,290199619787772728,5586346847185229596,108141141737193646020
%N A352383 G.f. A(x) satisfies: A(x) = (1 + 2*x*A(x)^3)^2 / (1 + x*A(x)^3).
%C A352383 Equals a bisection of A351771.
%C A352383 Self-convolution cube equals A352384.
%F A352383 G.f. A(x) satisfies:
%F A352383 (1) A(x) = (1 + 2*x*A(x)^3)^2 / (1 + x*A(x)^3).
%F A352383 (2) 0 = 4*A(x)^6*x^2 + (4 - A(x))*A(x)^3*x + (1 - A(x)).
%F A352383 (3) A(x) = 4 + 8*x*A(x)^3 - sqrt( A(x)^2 + 8*A(x) ).
%F A352383 (4) x = ( A(x) - 4 + sqrt( A(x)^2 + 8*A(x) ) ) / (8*A(x)^3).
%F A352383 (5) A( x*(1+x)^3/(1+2*x)^6 ) = (1+2*x)^2/(1+x).
%F A352383 (6) A(x)^3 = (1/x) * Series_Reversion(  x*(1+x)^3/(1+2*x)^6 ).
%F A352383 a(n) ~ sqrt((221 - 29*sqrt(13))/78) * 2^(3*n) * (587 - 143*sqrt(13))^n  / (sqrt(Pi) * n^(3/2) * 3^(3*n+1)). - _Vaclav Kotesovec_, Mar 15 2022
%e A352383 G.f.: A(x) = 1 + 3*x + 28*x^2 + 350*x^3 + 5020*x^4 + 78023*x^5 + 1278340*x^6 + 21740636*x^7 + 380161308*x^8 + 6792111260*x^9 + ...
%e A352383 where A(x) satisfies
%e A352383 A(x) = (1 + 2*x*A(x)^3)^2 / (1 + x*A(x)^3).
%e A352383 Related series.
%e A352383 The cube of the g.f. A(x) equals the g.f. of A352384:
%e A352383 A(x)^3 = 1 + 9*x + 111*x^2 + 1581*x^3 + 24468*x^4 + 399735*x^5 + 6784186*x^6 + 118444293*x^7 + ... + A352384(n)*x^n + ...
%e A352383 and equals (1/x) * Series_Reversion(  x*(1+x)^3/(1+2*x)^6 ).
%t A352383 CoefficientList[(InverseSeries[Series[x*(1 + x)^3/(1 + 2*x)^6, {x, 0, 20}], x]/x)^(1/3), x] (* _Vaclav Kotesovec_, Mar 15 2022 *)
%o A352383 (PARI) /* Using Series Reversion */
%o A352383 {a(n) = my(A = ((1/x)*serreverse( x*(1+x)^3/(1+2*x +x^2*O(x^n))^6 ))^(1/3)); polcoeff(A,n)}
%o A352383 for(n=0,30,print1(a(n),", "))
%Y A352383 Cf. A352384, A351771, A188110.
%K A352383 nonn
%O A352383 0,2
%A A352383 _Paul D. Hanna_, Mar 14 2022