%I #8 Nov 09 2018 18:23:33
%S 1,1,1,25,73,1025,4913,48985,311305,2393953,17903761,140986201,
%T 1096160649,7777051265,61667165361,597402170649,4836234935497,
%U 4245154618465,-25145215353455,12982383457107609,139920724294631369,-5479397854898810111,-68618853272591110863,3588130738987950942681,48514725864891831998601,-2781644195772240632990623
%N G.f. D(x) satisfies: Sum_{n>=0} n * (x + (-1)^n*A(x))^n = 0, where A(x) = D^8(x), the 8th iteration of D(x), and A(x) is the g.f. of A318640.
%C a(n) (mod 8) = 1.
%C a(n) (mod 16) has period 4 after initial term: [1, 1,1,9,9, 1,1,9,9, ...].
%H Paul D. Hanna, <a href="/A318643/b318643.txt">Table of n, a(n) for n = 1..400</a>
%F G.f. D(x) satisfies:
%F (1) D(-D(-x)) = x.
%F (2) 0 = Sum_{n>=0} (-1)^n * n * ( D(D(D(D(x)))) - (-1)^n*D(D(D(D(-x)) )^n.
%F (3) 0 = (A-x)*(1 + (A-x)^2)/(1 - (A-x)^2)^2 - 2*(A+x)^2/(1 - (A+x)^2)^2, where A = D^8(x), i.e., A(x) = D(D(D(D(D(D(D(D(x)))))))).
%e G.f.: 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 + ...
%e where D(-D(-x)) = x.
%e RELATED SERIES.
%e (a) If D(D(D(D( D(D(D(D(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 D(D(D(D(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 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 + ... + A318642(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(HALF(x*Ser(A)))),n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A318640, A318641, A318642.
%K sign
%O 1,4
%A _Paul D. Hanna_, Aug 31 2018