OFFSET
0,2
COMMENTS
Note that G(x) = Sum_{n>=0} x^n * (q^n + G(x))^n / (1 + q^n*x*G(x))^(n+1) holds at q = 1 when G(x) = 1/(1-x).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..100
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * (2^n + A(x))^n / (1 + 2^n*x*A(x))^(n+1).
(2) A(x) = Sum_{n>=0} x^n * (2^n - A(x))^n / (1 - 2^n*x*A(x))^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 14*x^2 + 456*x^3 + 62242*x^4 + 32889024*x^5 + 68215273298*x^6 + 561507179207912*x^7 + 18430985645896044938*x^8 + ...
such that
A(x) = 1/(1 + x*A(x)) + x*(2 + A(x))/(1 + 2*x*A(x))^2 + x^2*(2^2 + A(x))^2/(1 + 2^2*x*A(x))^3 + x^3*(2^3 + A(x))^3/(1 + 2^3*x*A(x))^3 + x^4*(2^4 + A(x))^4/(1 + 2^4*x*A(x))^4 + x^5*(2^5 + A(x))^5/(1 + 2^5*x*A(x))^6 + ...
also,
A(x) = 1/(1 - x*A(x)) + x*(2 - A(x))/(1 - 2*x*A(x))^2 + x^2*(2^2 - A(x))^2/(1 - 2^2*x*A(x))^3 + x^3*(2^3 - A(x))^3/(1 - 2^3*x*A(x))^3 + x^4*(2^4 - A(x))^4/(1 - 2^4*x*A(x))^4 + x^5*(2^5 - A(x))^5/(1 - 2^5*x*A(x))^6 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=Vec(sum(n=0, #A, x^n*(2^n + Ser(A))^n/(1 + 2^n*x*Ser(A))^(n+1)))); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 26 2019
STATUS
approved