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%I #7 Nov 09 2018 18:15:28
%S 1,2,4,56,304,2944,22592,196864,1700352,14416896,127798272,1141090304,
%T 10314440704,92351643648,836353277952,7885603078144,74487690493952,
%U 634585643614208,5307259488829440,71216920022482944,950665921628733440,-1672932555803852800,-180293500222111219712,4877207777660970729472,164125283521717784805376,-3434577882064669615587328
%N G.f. C(x) satisfies: Sum_{n>=0} (-1)^n * n * ( C(C(x)) - (-1)^n*C(C(-x)) )^n = 0.
%H Paul D. Hanna, <a href="/A318642/b318642.txt">Table of n, a(n) for n = 1..400</a>
%F G.f. C(x) satisfies:
%F (1) C(-C(-x)) = x.
%F (2) 0 = Sum_{n>=0} (-1)^n * n * ( C(C(x)) - (-1)^n*C(C(-x)) )^n.
%F (3) 0 = Sum_{n>=0} n * ( x + (-1)^n*C(C(C(C(x)))) )^n.
%F (4) 0 = (A-x)*(1 + (A-x)^2)/(1 - (A-x)^2)^2 - 2*(A+x)^2/(1 - (A+x)^2)^2, where A = C(C(C(C(x)))).
%F (5) C(x) = D(D(x)), the 2nd iteration of the g.f. D(x) of A318643.
%e G.f.: C(x) = x + 2*x^2 + 4*x^3 + 56*x^4 + 304*x^5 + 2944*x^6 + 22592*x^7 + 196864*x^8 + 1700352*x^9 + 14416896*x^10 + 127798272*x^11 + 1141090304*x^12 + ...
%e where C(-C(-x)) = x.
%e RELATED SERIES.
%e (a) If C(C(C(C(x)))) = A(x) then
%e A(x) = x + 8*x^2 + 64*x^3 + 704*x^4 + 8704*x^5 + 113536*x^6 + 1544192*x^7 + 21671936*x^8 + 311468032*x^9 + 4560963584*x^10 + ... + A318640(n)*x^n + ...
%e such that
%e 0 = (x - A(x)) + 2*(x + A(x))^2 + 3*(x - A(x))^3 + 4*(x + A(x))^4 + 5*(x - A(x))^5 + 6*(x + A(x))^6 + 7*(x - A(x))^7 + 8*(x + A(x))^8 + 9*(x - A(x))^9 + 10*(x + A(x))^10 + ...
%e (b) If C(C(x)) = B(x) then
%e B(x) = x + 4*x^2 + 16*x^3 + 160*x^4 + 1408*x^5 + 13760*x^6 + 140288*x^7 + 1459200*x^8 + 15595520*x^9 + 168584192*x^10 + 1847791616*x^11 + ... + A318641(n)*x^n + ...
%e such that
%e 0 = (B(x) + B(-x)) - 2*(B(x) - B(-x))^2 + 3*(B(x) + B(-x))^3 - 4*(B(x) - B(-x))^4 + 5*(B(x) + B(-x))^5 - 6*(B(x) - B(-x))^6 + 7*(B(x) + B(-x))^7 - 8*(B(x) - B(-x))^8 + 9*(B(x) + B(-x))^9 - 10*(B(x) - B(-x))^10 +- ...
%e (c) If D(D(x)) = C(x), then
%e D(x) = x + x^2 + x^3 + 25*x^4 + 73*x^5 + 1025*x^6 + 4913*x^7 + 48985*x^8 + 311305*x^9 + 2393953*x^10 + 17903761*x^11 + 140986201*x^12 + 1096160649*x^13 + ... + A318643(n)*x^n + ...
%e where D(-D(-x)) = x.
%o (PARI) {HALF(F) = my(H=x); for(i=1,#F, H = (H + subst(F,x,serreverse(H +x*O(x^#F))))/2);H}
%o {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, m*(x + (-1)^m*x*Ser(A))^m), #A)); polcoeff( HALF(HALF(x*Ser(A))),n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A318640, A318641, A318643.
%K sign
%O 1,2
%A _Paul D. Hanna_, Aug 31 2018
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