OFFSET
0,2
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/(1-x)^(3*n) - A(x) )^n.
(2) 1 = Sum_{n>=0} ( 1 - (1-x)^(3*n)*A(x) )^n / (1-x)^(3*n^2).
(3) 1 = Sum_{n>=0} (1-x)^(3*n) / ( (1-x)^(3*n) + A(x) )^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 3*x + 15*x^2 + 262*x^3 + 8616*x^4 + 384873*x^5 + 21181421*x^6 + 1372455324*x^7 + 101895990777*x^8 + 8511828635054*x^9 + 789539638329648*x^10 + ...
such that
1 = 1 + (1/(1-x)^3 - A(x)) + (1/(1-x)^6 - A(x))^2 + (1/(1-x)^9 - A(x))^3 + (1/(1-x)^12 - A(x))^4 + (1/(1-x)^15 - A(x))^5 + (1/(1-x)^18 - A(x))^6 + (1/(1-x)^21 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1-x)^3/((1-x)^3 + A(x))^2 + (1-x)^6/((1-x)^6 + A(x))^3 + (1-x)^9/((1-x)^9 + A(x))^4 + (1-x)^12/((1-x)^12 + A(x))^5 + (1-x)^15/((1-x)^15 + A(x))^6 + (1-x)^18/((1-x)^18 + A(x))^7 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1-x)^(-3*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 20 2019
STATUS
approved