A033469 Denominator of Bernoulli(2n,1/2).

1, 12, 240, 1344, 3840, 33792, 5591040, 49152, 16711680, 104595456, 173015040, 289406976, 22900899840, 201326592, 116769423360, 7689065201664, 1095216660480, 51539607552, 65942866278481920, 824633720832, 7438196161904640, 3971435999526912

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

From the von Staudt-Clausen theorem it follows that a(n) can be computed without using Bernoulli polynomials or the 'denominator'-function (see the Sage implementation). - Peter Luschny, Mar 24 2014

References

• J. R. Philip, The symmetrical Euler-Maclaurin summation formula, Math. Sci., 6, 1981, pp. 35-41.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..250
• Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Cf. A001896.

Programs

Maple

with(numtheory); seq(denom(bernoulli(2*n, 1/2)), n=0..20);

Mathematica

Table[ BernoulliB[2*n, 1/2] // Denominator, {n, 0, 18}] (* Jean-François Alcover, Apr 15 2013 *)
a[ n_] := If[ n < 0, 0, (2 n)! SeriesCoefficient[ x/2 / Sinh[x/2], {x, 0, 2 n}] // Denominator]; (* Michael Somos, Sep 21 2016 *)

PARI

a(n)=denominator(subst(bernpol(2*n,x),x,1/2)); \\ Joerg Arndt, Apr 17 2013

Sage

def A033469(n):
    if n == 0: return 1
    M = map(lambda i: i+1, divisors(2*n))
    return 2^(2*n-1)*mul(filter(lambda s: is_prime(s), M))
[A033469(n) for n in (0..21)] # Peter Luschny, Mar 24 2014

Formula

a(n) = denominator(2*(2*Pi)^(-2*n)*(2*n)!*Li_{2*n}(-1)). - Peter Luschny, Jun 29 2012

a(n) = A081294(n) * A002445(n) for n > 0. - Paul Curtz, Apr 17 2013

Apparently, denominators of the fractions with e.g.f. (x/2) / sinh(x/2). - Tom Copeland, Sep 17 2016

Extensions

More terms from Joerg Arndt, Apr 17 2013