A224783 Denominator of Bernoulli(n,1/2) - Bernoulli(n,0).

1, 2, 4, 1, 16, 1, 64, 1, 256, 1, 1024, 1, 4096, 1, 16384, 1, 65536, 1, 262144, 1, 1048576, 1, 4194304, 1, 16777216, 1, 67108864, 1, 268435456, 1, 1073741824, 1, 4294967296, 1, 17179869184, 1, 68719476736, 1, 274877906944, 1, 1099511627776

Offset: 0

Author

Paul Curtz, Apr 17 2013

Keywords

• nonn
• frac
• less
• easy

Comments

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

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.

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.

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..250
• Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

Crossrefs

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

Programs

Maple

A224783 := proc(n)
    bernoulli(n,1/2)-bernoulli(n) ;
    denom(%) ;
end proc: # R. J. Mathar, Apr 25 2013

Mathematica

Table[Denominator[BernoulliB[n, 1/2] - BernoulliB[n, 0]], {n, 0, 50}] (* Vincenzo Librandi, Mar 19 2014 *)

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

Formula

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

From Colin Barker, Mar 19 2014: (Start)

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

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

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