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%I #10 Aug 30 2018 12:53:08
%S 1,4,16,96,640,4480,32768,247552,1915904,15113216,121069568,982269952,
%T 8054767616,66651537408,555846729728,4667099709440,39420882583552,
%U 334730329587712,2855644020342784,24464860907569152,210393112063770624,1815582450311495680,15716738746688733184,136443973020569042944,1187652950607087534080,10362818914536380170240
%N G.f. A(x) satisfies: Sum_{n>=1} (x + (-1)^n*A(x))^n = 0.
%H Vaclav Kotesovec, <a href="/A317998/b317998.txt">Table of n, a(n) for n = 1..200</a>
%F G.f. A(x) satisfies:
%F (1) A(-A(-x)) = x.
%F (2a) 0 = Sum_{n>=1} (x + (-1)^n*A(x))^n.
%F (2b) 1 = 1/(1 - (x + A(x))^2) + (x - A(x))/(1 - (x - A(x))^2).
%F (3) 0 = x*(1-x)*(1+x)^2 - (1-x)^2*A(x) + (1 + x + 2*x^2)*A(x)^2 + A(x)^3 - A(x)^4.
%F (4) A(x) = F(F(x)) where Sum_{n>=1} (-1)^n * (F(x) - (-1)^n*F(-x))^n = 0, and F(x) is the g.f. of A317999.
%F (5a) A(x) = x + 4 * Series_Reversion( x/sqrt(1 + 4*x^2 - 16*x^4) - 2*x^2 )^2.
%F (5b) A( x/sqrt(1 + 4*x^2 - 16*x^4) - 2*x^2 ) = x/sqrt(1 + 4*x^2 - 16*x^4) + 2*x^2.
%F a(n) ~ sqrt((-3*r^2 - 4*r^3 + (1+s)^2 + 2*r*(1 - s + 2*s^2)) / (1 + r + 2*r^2 + 3*s - 6*s^2)) / (2*sqrt(Pi) * n^(3/2) * r^(n - 1/2)), where r = 0.10802903243795080465651783320471389432467958205527... and s = 0.28488427575880229259717798521096697574952476989148... are real roots of the system of equations s^2*(1 + r + 2*r^2 + s) + (1-r)*r*(1+r)^2 = (1-r)^2 * s + s^4, s*(2 + 2*r + 4*r^2 + 3*s) = (1-r)^2 + 4*s^3. - _Vaclav Kotesovec_, Aug 26 2018
%e O.g.f.: A(x) = x + 4*x^2 + 16*x^3 + 96*x^4 + 640*x^5 + 4480*x^6 + 32768*x^7 + 247552*x^8 + 1915904*x^9 + 15113216*x^10 + ...
%e such that
%e 0 = (x - A(x)) + (x + A(x))^2 + (x - A(x))^3 + (x + A(x))^4 + (x - A(x))^5 + (x + A(x))^6 + (x - A(x))^7 + (x + A(x))^8 + ...
%e RELATED SERIES.
%e G.f. A(x) = F(F(x)) where
%e F(x) = x + 2*x^2 + 4*x^3 + 24*x^4 + 112*x^5 + 608*x^6 + 3392*x^7 + 19456*x^8 + 114688*x^9 + 681984*x^10 + 4120576*x^11 + 25182208*x^12 + ... + A317999(n)*x^n + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, (x + (-1)^m*x*Ser(A))^m), #A)); A[n]}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A317999.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 21 2018
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