A376227 - OEIS

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

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OFFSET

0,2

LINKS

Table of n, a(n) for n=0..12.

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...

EXAMPLE

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,