A303537 Expansion of ((1 + 4*x)/(1 - 4*x))^(1/4).

1, 2, 2, 12, 22, 124, 276, 1496, 3686, 19436, 51068, 263720, 724860, 3681880, 10466920, 52450992, 153093254, 758495564, 2261603564, 11096526344, 33676743956, 163842737928, 504738342808, 2437418983888, 7605947276508, 36487283224952, 115140704639576

Offset: 0

Author

Seiichi Manyama, Apr 25 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..1000

Crossrefs

Cf. A063886, A081654, A151403, A303538.

Programs

Mathematica

CoefficientList[Series[Surd[(1+4x)/(1-4x),4],{x,0,40}],x] (* Harvey P. Dale, Jul 25 2021 *)

PARI

N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(1/4))

Formula

a(n) ~ 2^(2*n + 1/4) / (Gamma(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 26 2018

n*a(n) = 2*a(n-1) + 4^2*(n-2)*a(n-2) for n > 1.

G.f.: A(x)=F(x*G(x^2)), where F(x) is the g.f. for A063886, and G(x) is the g.f. for A151403. - Alexander Burstein, Nov 13 2023