OFFSET
1,4
COMMENTS
a(n) (mod 8) = 1.
a(n) (mod 16) has period 4 after initial term: [1, 1,1,9,9, 1,1,9,9, ...].
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..400
FORMULA
G.f. D(x) satisfies:
(1) D(-D(-x)) = x.
(2) 0 = Sum_{n>=0} (-1)^n * n * ( D(D(D(D(x)))) - (-1)^n*D(D(D(D(-x)) )^n.
(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)))))))).
EXAMPLE
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 + ...
where D(-D(-x)) = x.
RELATED SERIES.
(a) If D(D(D(D( D(D(D(D(x)))) )))) = A(x) then
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 + ...
such that
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 + ...
(b) If D(D(D(D(x)))) = B(x) then
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 + ...
such that
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 +- ...
(c) If D(D(x)) = C(x), then
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 + ...
where D(-D(-x)) = x.
PROG
(PARI) {HALF(F) = my(H=x); for(i=1, #F, H = (H + subst(F, x, serreverse(H +x*O(x^#F))))/2); H}
{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)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Aug 31 2018
STATUS
approved