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%I #12 Jul 08 2024 12:06:02
%S 1,2,20,304,5728,123680,2942016,75356544,2048446208,58517294080,
%T 1744472116224,53991571224576,1728111953805312,57027260271980544,
%U 1935586663121272832,67440373642584637440,2408328339125296824320,88029604711420113190912,3289877540493975587913728,125591805077248068782129152
%N Expansion of g.f. A(x) satisfying A(A(x)) - 4*A(A(A(x)))^2 = x.
%H Paul D. Hanna, <a href="/A373314/b373314.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F (1) x = A(A(x)) - 4*A(A(A(x)))^2.
%F (2) x = A( A(x) - 4*A(A(x))^2 ).
%F (3) x = A(A( x - 4*A(x)^2 )).
%F (4) A(x) = A(A(A( x - 4*A(x)^2 ))).
%F (5) A(x) = A(A(A(x))) - 4*A(A(A(A(x))))^2.
%e G.f.: A(x) = x + 2*x^2 + 20*x^3 + 304*x^4 + 5728*x^5 + 123680*x^6 + 2942016*x^7 + 75356544*x^8 + 2048446208*x^9 + 58517294080*x^10 + ...
%e where A(A(x)) - 4*A(A(A(x)))^2 = x.
%e RELATED SERIES.
%e A(A(x)) = x + 4*x^2 + 48*x^3 + 816*x^4 + 16704*x^5 + 385600*x^6 + 9705728*x^7 + 261167104*x^8 + ...
%e A(A(A(x))) = x + 6*x^2 + 84*x^3 + 1584*x^4 + 35168*x^5 + 869152*x^6 + 23222336*x^7 + 659257728*x^8 + ...
%e A(A(A(x)))^2 = x^2 + 12*x^3 + 204*x^4 + 4176*x^5 + 96400*x^6 + 2426432*x^7 + 65291776*x^8 + ...
%e Let B(x) be the series reversion of A(A(x)), B( A(A(x)) ) = x, then
%e B(x) = x - 4*A(x)^2 = x - 4*x^2 - 16*x^3 - 176*x^4 - 2752*x^5 - 52288*x^6 - 1129728*x^7 - 26801152*x^8 - ...
%o (PARI) /* Using x = A(A(x)) - 4*A(A(A(x)))^2 */
%o {a(n) = my(A = [0,1],A1,A2,A3); for(i=1,n, A = concat(A,0); A1 = Ser(A);
%o A2 = subst(A1,x,A1); A3 = subst(A1,x,A2);
%o A[#A] = (1/2)*polcoeff(x - A2 + 4*A3^2, #A-1)); A[n+1]}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A213591, A373315.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 08 2024
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