OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n / (2 - A(x)/(1-x)^n)^(n+1),
(2) A(x) = Sum_{n>=0} ( 1/(1-x)^n + A(x) )^n / (2 + A(x)/(1-x)^n)^(n+1).
a(n) ~ c * A317904^n * n^n / exp(n), where c = 0.47061136383707... - Vaclav Kotesovec, Aug 11 2021
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 243*x^4 + 5041*x^5 + 122793*x^6 + 3433557*x^7 + 108824679*x^8 + 3857180303*x^9 + 151189425233*x^10 + ...
such that A = A(x) satisfies
A(x) = 1/(2 - A) + (1/(1-x) - A)/(2 - A/(1-x))^2 + (1/(1-x)^2 - A)^2/(2 - A/(1-x)^2)^3 + (1/(1-x)^3 - A)^3/(2 - A/(1-x)^3)^4 + (1/(1-x)^4 - A)^4/(2 - A/(1-x)^4)^5 + (1/(1-x)^5 - A)^5/(2 - A/(1-x)^5)^6 + ...
Also,
A(x) = 1/(2 + A) + (1/(1-x) + A)/(2 + A/(1-x))^2 + (1/(1-x)^2 + A)^2/(2 + A/(1-x)^2)^3 + (1/(1-x)^3 + A)^3/(2 + A/(1-x)^3)^4 + (1/(1-x)^4 + A)^4/(2 + A/(1-x)^4)^5 + (1/(1-x)^5 + A)^5/(2 + A/(1-x)^5)^6 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( 1/(1-x)^m - Ser(A) )^m / (2 - Ser(A)/(1-x)^m)^(m+1) ) ) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 24 2019
STATUS
approved