|
%I #20 Sep 08 2022 08:45:05
%S 5,27,121,503,2037,8179,32753,131055,524269,2097131,8388585,33554407,
%T 134217701,536870883,2147483617,8589934559,34359738333,137438953435,
%U 549755813849,2199023255511,8796093022165,35184372088787
%N a(n) = 2^(2n+1)*Sum_{k=1..2*n} binomial(2n+1,k)*Bernoulli(k)/2^k.
%H Vincenzo Librandi, <a href="/A069993/b069993.txt">Table of n, a(n) for n = 1..500</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,4).
%F From _Rolf Pleisch_, Aug 09 2009: (Start)
%F a(n) = 2(4^n-n) - 1;
%F a(n) = 2*A024037(n) - 1.
%F (End)
%F From _Colin Barker_, May 30 2020: (Start)
%F G.f.: x*(5 - 3*x + 4*x^2) / ((1 - x)^2*(1 - 4*x)).
%F a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3) for n>3.
%F (End)
%t LinearRecurrence[{6,-9,4},{5,27,121},30] (* _Harvey P. Dale_, Jul 03 2021 *)
%o (PARI) for(n=1,30,print1(-2*4^n*sum(i=1,2*n+1,binomial(2*n+1,i)*bernfrac(i)/2^i),","))
%o (PARI) Vec(x*(5 - 3*x + 4*x^2) / ((1 - x)^2*(1 - 4*x)) + O(x^25)) \\ _Colin Barker_, May 30 2020
%o (Magma) [2*(4^n-n)-1: n in [1..30]]; // _Vincenzo Librandi_, Jul 02 2011
%Y Cf. A024037. - _Rolf Pleisch_, Aug 09 2009
%K easy,nonn
%O 1,1
%A _Benoit Cloitre_, May 01 2002
|
