OFFSET
0,3
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n*(A(x)^n + sqrt(2))^n / (1 + sqrt(2)*x*A(x)^n)^(n+1).
(2) A(x) = Sum_{n>=0} x^n*(A(x)^n - sqrt(2))^n / (1 - sqrt(2)*x*A(x)^n)^(n+1).
From Paul D. Hanna, Nov 05 2021: (Start)
(3) A(x) = Sum_{n>=0} x^n*Sum_{k=0..n} sqrt(2)^k * binomial(n,k) * (A(x)^n - sqrt(2)*A(x)^k)^(n-k).
(4) A(x) = Sum_{n>=0} x^n*Sum_{k=0..n} (-sqrt(2))^k * binomial(n,k) * (A(x)^n + sqrt(2)*A(x)^k)^(n-k). (End)
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 22*x^5 + 92*x^6 + 440*x^7 + 2281*x^8 + 12493*x^9 + 71765*x^10 + 430651*x^11 + 2693870*x^12 + ...
Let q = sqrt(2), then
A(x) = 1/(1 + q*x) + x*(A(x) + q)/(1 + q*x*A(x))^2 + x^2*(A(x)^2 + q)^2 / (1 + q*x*A(x)^2)^3 + x^3*(A(x)^3 + q)^3/(1 + q*x*A(x)^3)^4 + x^4*(A(x)^4 + q)^4 / (1 + q*x*A(x)^4)^5 + x^5*(A(x)^5 + q)^5/(1 + q*x*A(x)^5)^6 + ...
also
A(x) = 1/(1 - q*x) + x*(A(x) - q)/(1 - q*x*A(x))^2 + x^2*(A(x)^2 - q)^2 / (1 - q*x*A(x)^2)^3 + x^3*(A(x)^3 - q)^3/(1 - q*x*A(x)^3)^4 + x^4*(A(x)^4 - q)^4 / (1 - q*x*A(x)^4)^5 + x^5*(A(x)^5 - q)^5/(1 - q*x*A(x)^5)^6 + ...
PROG
(PARI) {a(n) = my(A=[1, 1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff( sum(n=0, #A+1, x^n*(Ser(A)^n + sqrt(2))^n/(1 + sqrt(2)*x*Ser(A)^n)^(n+1) ), #A-1)); round( polcoeff(Ser(A), n) )}
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 12 2019
STATUS
approved