OFFSET
0,2
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/(1-x)^(4*n) - A(x) )^n.
(2) 1 = Sum_{n>=0} ( 1 - (1-x)^(4*n)*A(x) )^n / (1-x)^(4*n^2).
(3) 1 = Sum_{n>=0} (1-x)^(4*n) / ( (1-x)^(4*n) + A(x) )^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 4*x + 26*x^2 + 612*x^3 + 26919*x^4 + 1603680*x^5 + 117660064*x^6 + 10162944112*x^7 + 1005838347950*x^8 + 112009295740916*x^9 + 13850874442895434*x^10 + ...
such that
1 = 1 + (1/(1-x)^4 - A(x)) + (1/(1-x)^8 - A(x))^2 + (1/(1-x)^12 - A(x))^3 + (1/(1-x)^16 - A(x))^4 + (1/(1-x)^20 - A(x))^5 + (1/(1-x)^24 - A(x))^6 + (1/(1-x)^28 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1-x)^4/((1-x)^4 + A(x))^2 + (1-x)^8/((1-x)^8 + A(x))^3 + (1-x)^12/((1-x)^12 + A(x))^4 + (1-x)^16/((1-x)^16 + A(x))^5 + (1-x)^20/((1-x)^20 + A(x))^6 + (1-x)^24/((1-x)^24 + 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)^(-4*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