A349835 Expansion of (1 + 4*x)/sqrt(1 - 4*x).

1, 6, 14, 44, 150, 532, 1932, 7128, 26598, 100100, 379236, 1444456, 5525884, 21217224, 81719000, 315583920, 1221550470, 4737927780, 18409560180, 71645805000, 279227584020, 1089643989720, 4257130461480, 16649826582480, 65181326593500, 255401021170152

Offset: 0

Author

Jianing Song, Dec 01 2021

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

• Wikipedia, Cauchy product

Crossrefs

Cf. A000984, A349834, A349846, A349847.

Programs

PARI

a(n) = if(n, binomial(2*(n-1),n-1) * (8 - 2/n), 1)

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)).