A267796 - OEIS

%I #64 Apr 17 2022 08:21:30

%S 4,128,3072,65536,1310720,25165824,469762048,8589934592,154618822656,

%T 2748779069440,48378511622144,844424930131968,14636698788954112,

%U 252201579132747776,4323455642275676160,73786976294838206464,1254378597012249509888,21250649172913403461632

%N a(n) = (n+1)*4^(2n+1).

%C The partial sums of A001246(n)/a(n) converge absolutely. This series is also the hypergeometric function 1/4 * 4F3(1/2,1/2,1,1;2,2,2;1). - _Ralf Steiner_, Feb 09 2016

%H Colin Barker, <a href="/A267796/b267796.txt">Table of n, a(n) for n = 0..800</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (32,-256).

%F a(n) = A013709(n)*(n+1).

%F From _Colin Barker_, Mar 23 2017: (Start)

%F G.f.: 4 / (1 - 16*x)^2.

%F a(n) = 32*a(n-1) - 256*a(n-2) for n>1. (End)

%F From _Amiram Eldar_, Apr 17 2022: (Start)

%F a(n) = A193132(n+1)/3.

%F Sum_{n>=0} 1/a(n) = 4*log(16/15).

%F Sum_{n>=0} (-1)^n/a(n) = 4*log(17/16). (End)

%e For n=3, a(3) = (3+1)*4^(2*3+1) = 4*4^7 = 65536.

%t Table[(n + 1) 4^(2 n + 1), {n, 0, 20}] (* _Vincenzo Librandi_, Feb 10 2016 *)

%o (PARI) a(n) = (n+1)*4^(2*n+1); \\ _Michel Marcus_, Jan 28 2016

%o (PARI) Vec(4 / (1 - 16*x)^2 + O(x^30)) \\ _Colin Barker_, Mar 23 2017

%o (Magma) [(n+1)*4^(2*n+1): n in [0..45]]; // _Vincenzo Librandi_, Feb 10 2016

%Y Cf. A001246, A013709, A193132, A267982.

%K nonn,easy

%O 0,1

%A _Ralf Steiner_, Jan 24 2016

%E More terms from _Michel Marcus_, Jan 28 2016