OFFSET
1,2
FORMULA
G.f. A(x) satisfies:
(1) A(-A(-x)) = x.
(2a) 0 = Sum_{n>=1} (-1)^n * (A(x) - (-1)^n*A(-x))^n.
(2b) 1 = 1/(1 - (A(x) - A(-x))^2) - (A(x) + A(-x))/(1 - (A(x) + A(-x))^2).
(3) 0 = x*(1-x)*(1+x)^2 - (1-x)^2*A(A(x)) + (1 + x + 2*x^2)*A(A(x))^2 + A(A(x))^3 - A(A(x))^4.
(4) A(A(x)) = G(x) such that Sum_{n>=1} (x + (-1)^n*G(x))^n = 0, where G(x) is the g.f. of A317998.
EXAMPLE
G.f. A(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 + ...
such that
0 = (A(x) + A(-x)) - (A(x) - A(-x))^2 + (A(x) + A(-x))^3 - (A(x) - A(-x))^4 + (A(x) + A(-x))^5 - (A(x) - A(-x))^6 + (A(x) + A(-x))^7 - (A(x) - A(-x))^8 + ...
Also,
0 = (x - A(A(x))) + (x + A(A(x)))^2 + (x - A(A(x)))^3 + (x + A(A(x)))^4 + (x - A(A(x)))^5 + (x + A(A(x)))^6 + (x - A(A(x)))^7 + (x + A(A(x)))^8 + ...
RELATED SERIES.
A(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 + ... + A317998(n)*x^n + ...
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 21 2018
STATUS
approved