cp's OEIS Frontend

Let b(n) = A349834(n)/4^n, {b(n)} = {1, 3/2, 11/8, 23/16, 179/128, 365/256, 1439/1024, ...}. Since A349834(n) >= 4^n, Sum_{n>=0} b(n) is divergent. Let c(n) = a(n)/(-4)^n, {c(n)} = {1, -3/2, 7/8, -11/16, 75/128, -133/256, 483/1024, ...}. Since |c(n)| ~ 2/sqrt(Pi*n) and |c(n+1)|/|c(n)| = ((4*n+3)*(2*n-1)) / ((4*n-1)*(2*n+2)) < 1, Sum_{n>=0} c(n) is conditionally convergent by Leibniz's criterion. Note that Sum_{n>=0} b(n)*x^n = sqrt(1 + x)/(1 - x), Sum_{n>=0} c(n)*x^n = (1 - x)/sqrt(1 + x), hence the Cauchy product of Sum_{n>=0} b(n) and Sum_{n>=0} c(n) is 1 + 0 + 0 + .... {b(n)} and {c(n)} serve as an example such that the Cauchy product of a divergent series and a conditionally convergent series can be absolutely convergent.

Sum_{n>=0} (a(n)/(-4)^n) is the Cauchy product of Sum_{n>=0} (-A349845(n)/8^n) with itself.

Let b(n) = -a(n)/(-8)^n, {b(n)} = {1, 3/4, -19/32, 85/128, -1741/2048, 9685/8192, -114375/65536, ...}, then Sum_{n>=0} b(n) is clearly divergent since Sum_{n>=0} a(n)*x^n has radius of convergence 1/16. Let c(n) = -A349846(n)/(-4)^n, {c(n)} = {1, 3/2, -5/8, 7/16, -45/128, 77/256, -273/1024, ...}, then Sum_{n>=1} c(n) is the Cauchy product of Sum_{n>=0} b(n) with itself. Since |c(n)| ~ 1/sqrt(Pi*n) and |c(n+1)|/|c(n)| = ((2*n-1)*(2*n+3)) / ((2*n+1)*(2*n+2)) < 1, Sum_{n>=0} c(n) is conditionally convergent by Leibniz's criterion. {b(n)} serves as an example such that the Cauchy product of a divergent series with itself can be conditionally convergent.