A349847 Expansion of (1 + 8*x) / sqrt(1 - 4*x).

1, 10, 22, 68, 230, 812, 2940, 10824, 40326, 151580, 573716, 2183480, 8347612, 32033848, 123321400, 476050320, 1842020550, 7142249340, 27743985060, 107946346200, 420608639220, 1641030105000, 6410161959240, 25066222437360, 98115049503900, 384391435902552

Offset: 0

Author

Jianing Song, Dec 01 2021

Keywords

• nonn
• easy

Comments

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

Examples

			a(1) = binomial(0,0) * (12 - 2/1) = 10;
a(2) = binomial(2,1) * (12 - 2/2) = 22;
a(3) = binomial(4,2) * (12 - 2/3) = 68;
a(4) = binomial(6,3) * (12 - 2/4) = 230.
		

Links

• Wikipedia, Cauchy product

Crossrefs

Cf. A000984, A349845, A349835, A349846.

Programs

Mathematica

CoefficientList[Series[(1+8x)/Sqrt[1-4x],{x,0,30}],x] (* Harvey P. Dale, Jun 08 2023 *)

PARI

a(n) = if(n, binomial(2*(n-1),n-1) * (12 - 2/n), 1)

Formula

For n > 0, a(n) = 8*binomial(2*(n-1),n-1) + binomial(2*n,n) = binomial(2*(n-1),n-1) * (12 - 2/n).

a(n) ~ 4^n * (3/sqrt(Pi*n)).