OFFSET
0,3
COMMENTS
What is the value of the related limit, as x approaches 1, of the series:
lim_{x->1} abs(1-x) * Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1) = 0.83810457748...
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..4100
EXAMPLE
G.f.: A(x) = 1 - x + 3*x^2 - 5*x^3 + 4*x^4 - 2*x^5 + 9*x^6 - 23*x^7 + 22*x^8 - 8*x^9 + 12*x^10 - 24*x^11 + 27*x^12 - 67*x^13 + 128*x^14 - 112*x^15 + ...
where
A(x) = (1-x) * [1/(1+x) + x*(1 + x)/(1 + x^2)^2 + x^2*(1 + x^2)^2/(1 + x^3)^3 + x^3*(1 + x^3)^3/(1 + x^4)^4 + x^4*(1 + x^4)^4/(1 + x^5)^5 + x^5*(1 + x^5)^5/(1 + x^6)^6 + ...].
PROG
(PARI) {a(n) = my(A = (1-x)*sum(m=0, n, x^m * (1 + x^m +x*O(x^n) )^m / (1 + x^(m+1) +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, May 23 2022
STATUS
approved