A304941 Expansion of ((1 + 4*x)/(1 - 4*x))^(3/4).

1, 6, 18, 68, 246, 948, 3572, 13896, 53286, 208452, 807132, 3169080, 12346300, 48602760, 190150440, 750018448, 2943363078, 11627329764, 45736940364, 180897649368, 712881236052, 2822389182104, 11138924119512, 44137230865392, 174405194802524, 691557285091176

Offset: 0

Author

Seiichi Manyama, May 22 2018

Keywords

• nonn

Comments

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

((1 + 4*x)/(1 - 4*x))^(m/4): A303537 (m=1), A304940 (m=2), this sequence (m=3), A081654 (m=4).

Programs

Magma

[n le 2 select 6^(n-1) else 2*(3*Self(n-1) + 8*(n-3)*Self(n-2))/(n-1): n in [1..40]]; // G. C. Greubel, Jun 07 2023

Mathematica

CoefficientList[Series[((1+4x)/(1-4x))^(3/4),{x,0,30}],x] (* Harvey P. Dale, Oct 24 2020 *)

PARI

N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(3/4))

SageMath

@CachedFunction
def a(n): # a = A304941
    if n<2: return 6^n
    else: return 2*(3*a(n-1) + 8*(n-2)*a(n-2))//n
[a(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

Formula

n*a(n) = 6*a(n-1) + 4^2*(n-2)*a(n-2) for n > 1.

a(n) ~ 2^(2*n + 3/4) / (Gamma(3/4) * n^(1/4)). - Vaclav Kotesovec, May 28 2018