A224783 - OEIS

%I #32 Feb 12 2024 13:29:09

%S 1,2,4,1,16,1,64,1,256,1,1024,1,4096,1,16384,1,65536,1,262144,1,

%T 1048576,1,4194304,1,16777216,1,67108864,1,268435456,1,1073741824,1,

%U 4294967296,1,17179869184,1,68719476736,1,274877906944,1,1099511627776

%N Denominator of Bernoulli(n,1/2) - Bernoulli(n,0).

%C See A157779 and A157780 for values of Bernoulli(n,1/2), and A027641 and A027642 for values of Bernoulli(n,0).

%C B(n,1/2) - B(n,0) = 0, 1/2, -1/4, 0, 1/16, 0, -3/64, 0, 17/256, 0, -155/1024, 0, 2073/4096, 0, -38227/16384,... for n>=0.

%C The sequence of numerators is 0, 1, -1, 0, 1, 0, -3, 0, 17, 0, -155, 0, 2073, 0, -38227, 0, 929569, 0, -28820619, 0, 1109652905,...and appears to contain a mix of A001469 and A036968.

%H Vincenzo Librandi, <a href="/A224783/b224783.txt">Table of n, a(n) for n = 0..250</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-4).

%F a(n) = A059222(n+1) if n <> 1.

%F From _Colin Barker_, Mar 19 2014: (Start)

%F G.f.: (4*x^5-9*x^3-x^2+2*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)).

%F a(n) = 5*a(n-2)-4*a(n-4) for n>5.

%F a(n) = (1+(-2)^n-(-1)^n+2^n)/2 for n>1. (End).

%e a(0) = 1-1, a(1) = 0+1/2, a(2) = -1/12-1/6=-1/4.

%p A224783 := proc(n)

%p     bernoulli(n,1/2)-bernoulli(n) ;

%p     denom(%) ;

%p end proc: # _R. J. Mathar_, Apr 25 2013

%t Table[Denominator[BernoulliB[n, 1/2] - BernoulliB[n, 0]], {n, 0, 50}] (* _Vincenzo Librandi_, Mar 19 2014 *)

%o (PARI) Vec((4*x^5-9*x^3-x^2+2*x+1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)) + O(x^100)) \\ _Colin Barker_, Mar 20 2014

%Y Cf. A001469, A027641, A027642, A036968, A059222, A157779, A157780.

%K nonn,frac,less,easy

%O 0,2

%A _Paul Curtz_, Apr 17 2013