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%I #47 Apr 28 2022 20:20:55
%S 1,2,4,14,52,204,840,3574,15588,69332,313272,1433964,6635400,30988312,
%T 145871248,691403686,3296979524,15805913476,76135613784,368304184900,
%U 1788518253080,8715477003688,42605364060656,208878870197436,1026781984000680,5059692979338824,24989145569112880,123676728224877464,613295203581498768,3046761116509464624
%N O.g.f. A(x) satisfies: Sum_{n>=1} (x + (-1)^n*A(x))^n / n = 0.
%H Vaclav Kotesovec, <a href="/A316363/b316363.txt">Table of n, a(n) for n = 1..200</a>
%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
%F (1a) 0 = Sum_{n>=1} (x + (-1)^n*A(x))^n / n,
%F (1b) 0 = arctanh(x - A(x)) - log(1 - (x + A(x))^2)/2,
%F (1c) 1 - (x + A(x))^2 = (1+x - A(x))/(1-x + A(x)).
%F (2a) A(x) = x + (x + A(x))^2/(2 - (x + A(x))^2).
%F (2b) 0 = (2*x + x^2 - x^3) - (2 - 2*x + x^2)*A(x) + (1+x)*A(x)^2 + A(x)^3.
%F (3) A(-A(-x)) = x.
%F (4a) A(x) = -x + 2 * Series_Reversion( x - x^2/(1 - 2*x^2) ).
%F (4b) A(x) = x + 2 * Series_Reversion( x/sqrt(1 + 2*x^2) - x^2 )^2.
%F (5) (A'(x) - 1)/(A'(x) + 1) = (x + A(x))*(1 - (x - A(x))^2) / (1 - (x + A(x))^2). - _Paul D. Hanna_, Apr 28 2022
%F a(n) ~ sqrt(r*(r*(1-r) + s*(1+s)) / (1 + r + 3*s)) / (sqrt(Pi) * n^(3/2) * r^n), where r = 0.1912388335306640951515262439910852999016888421453... and s = 0.444963791747610196027930141875385171928290741217... are real roots of the system of equations r^2*(-1 + r + s) = s*(-2 + s + s^2) + r*(2 + 2*s + s^2), 2*r*(1 + s) + s*(2 + 3*s) = 2 + r^2. - _Vaclav Kotesovec_, Jul 06 2018
%e G.f.: A(x) = x + 2*x^2 + 4*x^3 + 14*x^4 + 52*x^5 + 204*x^6 + 840*x^7 + 3574*x^8 + 15588*x^9 + 69332*x^10 + 313272*x^11 + 1433964*x^12 + ...
%e such that
%e 0 = (x - A(x)) + (x + A(x))^2/2 + (x - A(x))^3/3 + (x + A(x))^4/4 + (x - A(x))^5/5 + (x + A(x))^6/6 + (x - A(x))^7/7 + (x + A(x))^8/8 + ...
%e thus arctanh(x - A(x)) - log(1 - (x + A(x))^2)/2 = 0
%e so that (1+x - A(x))/(1-x + A(x)) = 1 - (x + A(x))^2.
%e RELATED SERIES.
%e Series_Reversion(A(x)) = x - 2*x^2 + 4*x^3 - 14*x^4 + 52*x^5 - 204*x^6 + 840*x^7 - 3574*x^8 + ... + (-1)^(n-1)*a(n)*x^n + ...
%e Series_Reversion(x + A(x)) = 1/2*x - 1/4*x^2 - 1/8*x^4 - 1/16*x^6 - 1/32*x^8 - 1/64*x^10 - 1/128*x^12 - 1/256*x^14 + ... = x*(1 - x)*(2 + x)/(4 - 2*x^2); equivalently, Series_Reversion((x + A(x))/2) = x - x^2/(1 - 2*x^2).
%e Let F(x) be the g.f. of A317800, then A(x) = F(F(x)), where
%e F(x) = x + x^2 + x^3 + 4*x^4 + 10*x^5 + 33*x^6 + 105*x^7 + 354*x^8 + 1214*x^9 + 4206*x^10 + ... + A317800(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/m), #A)); A[n]}
%o for(n=1,30, print1(a(n),", "))
%Y Cf. A317800.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 03 2018
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