A376227 a(n) = Product_{k=1..n} (8^k - 1)/(2^k - 1) for n >= 1 with a(0) = 1.

1, 7, 147, 10731, 2929563, 3096548091, 12884736606651, 212765655585627963, 13998490777945220569659, 3676801592262757799164923963, 3859174628040582848761303356488763, 16194459027901983959148041623911690081339, 271764285812898926139442499827890355613945218107

Offset: 0

Author

Paul D. Hanna, Oct 09 2024

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 7*x + 147*x^2 + 10731*x^3 + 2929563*x^4 + 3096548091*x^5 + 12884736606651*x^6 + 212765655585627963*x^7 + ...
where the coefficients a(n) of x^n begin
a(0) = 1,
a(1) = 1 * 7,
a(2) = 1 * 7 * 21,
a(3) = 1 * 7 * 21 * 73,
a(4) = 1 * 7 * 21 * 73 * 273,
a(5) = 1 * 7 * 21 * 73 * 273 * 1057,
...
		

Crossrefs

Cf. A376530, A001576, A135576.

Programs

PARI

{a(n) = (1/3) * prod(k=0,n, 1 + 2^k + 2^(2*k))}
for(n=0,12,print1(a(n),", "))

Formula

G.f. A(x) = 1/(1 - 7*x/(1 + 7*x - 21*x/(1 + 21*x - 73*x/(1 + 73*x - 273*x/(1 + 273*x - 1057*x/(1 + 1057*x - 4161*x/(1 + ...))))))), a continued fraction.

a(n) = Product_{k=1..n} (1 + 2^k + 2^(2*k)) for n >= 1 with a(0) = 1.

a(n) = 2^(n*(n+1)/2) * Product_{k=1..n} (1/2^k + 1 + 2^k) for n >= 1.

a(n) ~ c * 2^(n*(n+1)) where c = Product_{n>=1} (1 + 1/2^n + 1/4^n) = 2.975905201850451176749639540825805061981174...