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%I
%S 1,1,1,4,10,33,105,354,1214,4206,14846,52750,189516,686745,2506913,
%T 9211226,34036230,126426446,471769950,1767460752,6645539212,
%U 25076120890,94937019050,360268374124,1369645176012,5226326126048,20039843858208,76654036799842,290534140464144,1123489897863753,4582416833711249,17212665701732282,45565498032190230
%N G.f. A(x) satisfies: Sum_{n>=1} ( A(x) - (-1)^n * A(-x) )^n * (-1)^n / n = 0.
%C Odd terms occur at a(2^k - 1) and a(2^k - 2) for k > 1 and at a(1), while a(n) is even elsewhere (conjecture).
%C First negative term is a(37).
%H Paul D. Hanna, <a href="/A317800/b317800.txt">Table of n, a(n) for n = 1..520</a>
%F G.f. A(x) satisfies:
%F (1) A(-A(-x)) = x.
%F (2a) Sum_{n>=1} ( A(x) - (-1)^n * A(-x) )^n * (-1)^n / n = 0.
%F (2b) A(A(x)) = B(x) such that Sum_{n>=1} ( x + (-1)^n * B(x) )^n / n = 0, where B(x) is the o.g.f. of A316363.
%F (3a) A(A(x)) = -x + 2 * Series_Reversion( x - x^2/(1 - 2*x^2) ).
%F (3b) A(A(x)) = x + 2 * Series_Reversion( x/sqrt(1 + 2*x^2) - x^2 )^2.
%F Let C = (A(x) + A(-x))/2 and S = (A(x) - A(-x))/2, then
%F (4a) arctanh(2*C) + log(1 - 4*S^2)/2 = 0,
%F (4b) 1 - 4*S^2 = (1 - 2*C)/(1 + 2*C),
%F (5a) S^2 = C/(1 + 2*C),
%F (5b) C = S^2/(1 - 2*S^2),
%F (6a) A(x) = S + S^2/(1 - 2*S^2),
%F (6b) A(x) = C + sqrt(C/(1 + 2*C)).
%F (7) 0 = (2*y + y^2 - y^3) - (2 - 2*y + y^2)*A(x) + (1 + y)*A(x)^2 + A(x)^3, where y = -A(-x) = Series_Reversion(A(x)).
%e G.f. A(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 + ...
%e Let the series bisections of g.f. A(x) be denoted by
%e C = (A(x) + A(-x))/2 = x^2 + 4*x^4 + 33*x^6 + 354*x^8 + 4206*x^10 + ...
%e S = (A(x) - A(-x))/2 = x + x^3 + 10*x^5 + 105*x^7 + 1214*x^9 + 14846*x^11 + ...
%e then from the definition we have
%e 0 = (2*C) - (2*S)^2/2 + (2*C)^3/3 - (2*S)^4/4 + (2*C)^5/5 - (2*S)^6/6 + (2*C)^7/7 - (2*S)^8/8 + ...
%e thus arctanh(2*C) + log(1 - 4*S^2)/2 = 0,
%e so that (1 - 2*C)/(1 + 2*C) = 1 - 4*S^2.
%e RELATED SERIES.
%e A(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 + ... + A316363(n)*x^n + ...
%e where A(A(x)) = -x + 2 * Series_Reversion( x - x^2/(1 - 2*x^2) ).
%o (PARI) /* From: A(x) = S + S^2/(1 - 2*S^2) and A(x) = Series_Reversion(-A(-x)) */
%o {a(n) = my(A=[1,1],S); for(i=1,n, S=(x*Ser(A) - subst(x*Ser(A),x,-x))/2; A=concat(Vec( S + S^2/(1 - 2*S^2) ),0); if(#A%2==1,A = (A + Vec( serreverse(subst(-x*Ser(A),x,-x)) ) )/2 ); );A[n]}
%o for(n=1,30, print1(a(n),", "))
%Y Cf. A316363.
%K sign
%O 1,4
%A _Paul D. Hanna_, Aug 09 2018
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