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%I #9 Dec 31 2021 12:19:11
%S 1,-1,-1,2,-2,3,-1,-7,10,2,-15,2,34,-51,17,73,-218,323,-135,-467,1139,
%T -1279,430,1587,-4274,5798,-3249,-5664,19061,-26208,9976,34430,-77516,
%U 62473,45193,-186383,173814,186306,-747220,754744,678009,-3221980,4339761,-491173,-8988984,17693649,-12889827,-13658278,48487713
%N G.f. A(x) satisfies: A(x) = A(x^2 - x^3)/x.
%C Equals the series reversion of the g.f. of A273162.
%H Paul D. Hanna, <a href="/A273218/b273218.txt">Table of n, a(n) for n = 1..1030</a>
%F G.f.: A(x) = Product_{n>=0} F(n), where F(0) = x, F(1) = 1-x, and F(n+1) = 1 - (1 - F(n))^2 * F(n) for n > 0. - _Paul D. Hanna_, Dec 30 2021
%e G.f.: A(x) = x - x^2 - x^3 + 2*x^4 - 2*x^5 + 3*x^6 - x^7 - 7*x^8 + 10*x^9 + 2*x^10 - 15*x^11 + 2*x^12 + 34*x^13 - 51*x^14 + 17*x^15 + 73*x^16 - 218*x^17 +...
%e such that A(x) = A(x^2 - x^3)/x.
%e RELATED SERIES.
%e Let B(x) be the series reversion of g.f. A(x), so that B(A(x)) = x, then
%e B(x) = x + x^2 + 3*x^3 + 8*x^4 + 28*x^5 + 95*x^6 + 351*x^7 + 1309*x^8 + 5056*x^9 + 19787*x^10 + 78847*x^11 +...+ A273162(n)*x^n +...
%e such that B(x*A(x)) = x^2 - x^3.
%e From _Paul D. Hanna_, Dec 30 2021: (Start)
%e GENERATING METHOD.
%e Define F(n), a polynomial in x of order 3^(n-1), by the following recurrence:
%e F(0) = x,
%e F(1) = (1 - x),
%e F(2) = (1 - x^2 * (1-x)),
%e F(3) = (1 - x^4 * (1-x)^2 * F(2)),
%e F(4) = (1 - x^8 * (1-x)^4 * F(2)^2 * F(3)),
%e F(5) = (1 - x^16 * (1-x)^8 * F(2)^4 * F(3)^2 * F(4)),
%e ...
%e F(n+1) = 1 - (1 - F(n))^2 * F(n)
%e ...
%e Then the g.f. A(x) equals the infinite product:
%e A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
%e that is,
%e A(x) = x * (1-x) * (1 - x^2*(1-x)) * (1 - x^4*(1-x)^2*(1 - x^2*(1-x))) * (1 - x^8*(1-x)^4*(1 - x^2*(1-x))^2*(1 - x^4*(1-x)^2*(1 - x^2*(1-x)))) * ...
%e (End)
%e From _Paul D. Hanna_, Dec 31 2021: (Start)
%e SPECIFIC VALUES.
%e The infinite product formula allows us to evaluate the function A(x) at certain x rather quickly.
%e A(1/2) = (1/2) * (1/2) * (7/2^3) * (505/2^9) * (134192983/2^27) * (2417851557060878608942777/2^81) * ... = 0.21571949163622469813172568...
%e A(2/3) = (2/3) * (1/3) * (23/3^3) * (19315/3^9) * (7622981770427/3^27) * ... = 0.18569744603728983530046038...
%e A(1/3) = (1/3) * (2/3) * (25/3^3) * (19583/3^9) * (7625401654987/3^27) * ... = 0.20471068371640502928595863...
%e The first relative maximum value of A(x) is given by
%e A(0.45249959935125940...) = 0.21748935249823157...
%e (End)
%o (PARI) {a(n) = my(A=x); for(i=1, #binary(n)+1, A = subst(A,x, x^2 - x^3 + x^2*O(x^n))/x); polcoeff(A,n) }
%o for(n=1,60,print1(a(n),", "))
%o (PARI) /* Using Infinite Product Formula */
%o {F(n) = my(G=x); if(n==0,G=x, if(n==1,G=1-x, G = 1 - (1 - F(n-1))^2*F(n-1) ));G}
%o {a(n) = my(A = prod(k=0,#binary(n), F(k) +x*O(x^n))); polcoeff(A,n)}
%o for(n=1,50,print1(a(n),", ")) \\ _Paul D. Hanna_, Dec 30 2021
%Y Cf. A273162, A350432.
%K sign
%O 1,4
%A _Paul D. Hanna_, May 17 2016
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