OFFSET
0,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 1 + x = Sum_{n>=0} x^n*(A(x)^n + 1)^n/(1 + x*A(x)^n)^(n+1).
(2) 1 + x = Sum_{n>=0} x^n*(A(x)^n - 1)^n/(1 - x*A(x)^n)^(n+1).
EXAMPLE
G.f.: A(x) = 1 - x + x^2 - 6*x^3 + 10*x^4 - 27*x^5 + 28*x^6 - 107*x^7 + 502*x^8 - 1996*x^9 - 1015*x^10 + 39035*x^11 - 76739*x^12 - 1078632*x^13 + ...
such that
1 + x = 1/(1 + x*A(x)) + x*(A(x) + 1)/(1 + x*A(x))^2 + x^2*(A(x)^2 + 1)^2 / (1 + x*A(x)^2)^3 + x^3*(A(x)^3 + 1)^3/(1 + x*A(x)^3)^4 + x^4*(A(x)^4 + 1)^4 / (1 + x*A(x)^4)^5 + x^5*(A(x)^5 + 1)^5 / (1 + x*A(x)^5)^6 + ...
also
1 + x = 1/(1 - x*A(x)) + x*(A(x) - 1)/(1 - x*A(x))^2 + x^2*(A(x)^2 - 1)^2 / (1 - x*A(x)^2)^3 + x^3*(A(x)^3 - 1)^3/(1 - x*A(x)^3)^4 + x^4*(A(x)^4 - 1)^4 / (1 - x*A(x)^4)^5 + x^5*(A(x)^5 - 1)^5 / (1 - x*A(x)^5)^6 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A+1, x^n*(Ser(A)^n + 1)^n/(1 + x*Ser(A)^n)^(n+1) ), #A)); polcoeff(Ser(A), n)}
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Mar 12 2019
STATUS
approved