login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A305031 Expansion of ((1 + 2*x)/(1 - 2*x))^(3/2). 3

%I #26 Jun 07 2023 20:05:51

%S 1,6,18,44,102,228,500,1080,2310,4900,10332,21672,45276,94248,195624,

%T 404976,836550,1724580,3549260,7293000,14965236,30669496,62783448,

%U 128388624,262303132,535422888,1092063000,2225728400,4533175800,9226818000,18769219920,38158909920

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

%C 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.

%H Seiichi Manyama, <a href="/A305031/b305031.txt">Table of n, a(n) for n = 0..3000</a>

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

%F a(n) ~ 2^(n + 5/2) * sqrt(n/Pi). - _Vaclav Kotesovec_, May 28 2018

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

%o (PARI) N=66; x='x+O('x^N); Vec(((1+2*x)/(1-2*x))^(3/2))

%o (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

%o (SageMath)

%o @CachedFunction

%o def a(n): # b = A305031

%o if n<2: return 6^n

%o else: return 2*(3*a(n-1) + 2*(n-2)*a(n-2))//n

%o [a(n) for n in range(41)] # _G. C. Greubel_, Jun 07 2023

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

%Y Cf. A304941, A305032.

%K nonn

%O 0,2

%A _Seiichi Manyama_, May 24 2018

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 05:48 EDT 2024. Contains 371265 sequences. (Running on oeis4.)