A305031 Expansion of ((1 + 2*x)/(1 - 2*x))^(3/2).

1, 6, 18, 44, 102, 228, 500, 1080, 2310, 4900, 10332, 21672, 45276, 94248, 195624, 404976, 836550, 1724580, 3549260, 7293000, 14965236, 30669496, 62783448, 128388624, 262303132, 535422888, 1092063000, 2225728400, 4533175800, 9226818000, 18769219920, 38158909920

Offset

0,2

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..3000

Formula

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

a(n) ~ 2^(n + 5/2) * sqrt(n/Pi). - Vaclav Kotesovec, May 28 2018

Mathematica

CoefficientList[Series[((1+2*x)/(1-2*x))^(3/2), {x, 0, 40}], x] (* G. C. Greubel, Jun 07 2023 *)

Prog

(PARI) N=66; x='x+O('x^N); Vec(((1+2*x)/(1-2*x))^(3/2))
(Magma) [n le 2 select 6^(n-1) else 2*(3*Self(n-1) + 2*(n-3)*Self(n-2))/(n-1): n in [1..40]]; // G. C. Greubel, Jun 07 2023
(SageMath)
@CachedFunction
def a(n): # b = A305031
    if n<2: return 6^n
    else: return 2*(3*a(n-1) + 2*(n-2)*a(n-2))//n
[a(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

Crossrefs

((1 + 2*x)/(1 - 2*x))^(m/2): A063886 (m=1), this sequence (m=3), A241204 (m=4).

Cf. A304941, A305032.

Sequence in context: A303649 A095170 A261647 * A031128 A304161 A261016

Adjacent sequences: A305028 A305029 A305030 * A305032 A305033 A305034

Keyword

nonn

Author

Seiichi Manyama, May 24 2018

Status

approved