OFFSET
0,2
COMMENTS
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.
LINKS
Wikipedia, Cauchy product
FORMULA
For n > 0, a(n) = binomial(2*n,n) + 4*binomial(2*(n-1),n-1) = binomial(2*(n-1),n-1) * (8 - 2/n).
a(n) ~ 4^n * (2/sqrt(Pi*n)).
EXAMPLE
a(1) = binomial(0,0) * (8 - 2/1) = 6;
a(2) = binomial(2,1) * (8 - 2/2) = 14;
a(3) = binomial(4,2) * (8 - 2/3) = 44;
a(4) = binomial(6,3) * (8 - 2/4) = 150.
PROG
(PARI) a(n) = if(n, binomial(2*(n-1), n-1) * (8 - 2/n), 1)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Dec 01 2021
STATUS
approved