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A224783 Denominator of Bernoulli(n,1/2) - Bernoulli(n,0). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

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).

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).

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A184176 A163546 A172385 * A214058 A173820 A030043

Adjacent sequences:  A224780 A224781 A224782 * A224784 A224785 A224786

KEYWORD

nonn,frac,less,easy

AUTHOR

Paul Curtz, Apr 17 2013

STATUS

approved

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Last modified August 27 17:19 EDT 2015. Contains 261095 sequences.