OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..50
FORMULA
Let q = sqrt(128/3), then g.f. A(x) satisfies:
(1) 1 + 4*x = Sum_{n>=0} (4^n + q * A(x))^n * x^n / (1 + 4^n * q * x*A(x))^(n+1).
(2) 1 + 4*x = Sum_{n>=0} (4^n - q * A(x))^n * x^n / (1 - 4^n * q * x*A(x))^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 416*x + 8029248*x^2 + 2188617320448*x^3 + 9219890831036553216*x^4 + 618951997873353332851408896*x^5 + 664612512289053409746943478501867520*x^6 + ...
Let q = sqrt(128/3), then
1 + 4*x = 1/(1+x*q*A(x)) + (4 + q*A(x))*x/(1 + 4*x*q*A(x))^2 + (4^2 + q*A(x))^2*x^2/(1 + 4^2*x*q*A(x))^3 + (4^3 + q*A(x))^3*x^3/(1 + 4^3*x*q*A(x))^4 + (4^4 + q*A(x))^4*x^4/(1 + 4^4*x*q*A(x))^5 + (4^5 + q*A(x))^5*x^5/(1 + 4^5*x*q*A(x))^6 + (4^6 + q*A(x))^6*x^6/(1 + 4^6*x*q*A(x))^7 + ...
and also
1 + 4*x = 1/(1-x*q*A(x)) + (4 - q*A(x))*x/(1 - 4*x*q*A(x))^2 + (4^2 - q*A(x))^2*x^2/(1 - 4^2*x*q*A(x))^3 + (4^3 - q*A(x))^3*x^3/(1 - 4^3*x*q*A(x))^4 + (4^4 - q*A(x))^4*x^4/(1 - 4^4*x*q*A(x))^5 + (4^5 - q*A(x))^5*x^5/(1 - 4^5*x*q*A(x))^6 + (4^6 - q*A(x))^6*x^6/(1 - 4^6*x*q*A(x))^7 + ...
PROG
(PARI) /* Requires high precision */
{a(n) = my(q=sqrt(128/3), A=[1, 416, 0]); for(i=0, n,
A=concat(A, 0); A[#A-1] = round( polcoeff( sum(n=0, #A, (4^n + q * Ser(A))^n * x^n / (1 + 4^n * q * x*Ser(A))^(n+1) ), #A)/512)); A[n+1]}
for(n=0, 10, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 16 2019
STATUS
approved