A304915 Expansion of ((1 + 16*x)/(1 - 16*x))^(1/16).

1, 2, 2, 172, 342, 26556, 67220, 4875160, 14125030, 973837420, 3087573628, 204536051176, 692771715836, 44412235657176, 158358513025896, 9874709152875568, 36706645561910150, 2234840966950941260, 8601116786415880940, 512801585354912006600, 2032977466125710169236

Offset: 0

Author

Seiichi Manyama, May 21 2018

Keywords

• nonn

Comments

Let ((1 + k*x)/(1 - k*x))^(1/k) = a(0) + a(1)*x + a(2)*x^2 + ...

Then n*a(n) = 2*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A063886, A303537, A303538.

Programs

Mathematica

CoefficientList[Series[((1+16x)/(1-16x))^(1/16),{x,0,30}],x] (* Harvey P. Dale, Jul 21 2021 *)

PARI

N=66; x='x+O('x^N); Vec(((1+16*x)/(1-16*x))^(1/16))

Formula

n*a(n) = 2*a(n-1) + 16^2*(n-2)*a(n-2) for n > 1.

a(n) ~ 2^(4*n + 1/16) / (Gamma(1/16) * n^(15/16)) * (1 - (-1)^n * sqrt(2 - sqrt(2 + sqrt(2))) * 2^(7/8) * Gamma(1/16)^2 / (64*Pi*n^(1/8))). - Vaclav Kotesovec, May 21 2018