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A216912 a(n) = denominator(B°(2*n))/4 where the B°(n) are Zagier's modified Bernoulli numbers. 1
6, 20, 315, 280, 66, 3003, 78, 9520, 305235, 20900, 138, 19734, 6, 7540, 15575175, 590240, 6, 107666559, 222, 11996600, 50536395, 19780, 282, 31534932, 66, 1060, 48532365, 738920, 354, 83912718435, 366, 1180480, 485415, 1340, 60918, 3667092237666, 438, 740 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Sequence given for a(1)-a(15) in Note 6.2, p.13 of Dixit and others. Let alpha(n) = denominator(B°(n)). The data suggests that the prime factors of alpha(2n) are bounded by 2n + 1. The sequence alpha(n) is periodic, with minimal period p, if and only if its generating function A(z) = Sum_{n>=0} g(n)*z^n is a rational function of z such that, when written in reduced form, the denominator has the form D(z) = 1 - z^p. (Lemma 12.1, p. 24.)

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..500

M. W. Coffey, V. de Angelis, A. Dixit, V. H. Moll, et al., The Zagier polynomials. Part II: Arithmetic properties of coefficients, arXiv:1303.6590 [math.NT], 2013.

A. Dixit, V. H. Moll, Ch. Vignat, The Zagier modification of Bernoulli numbers and a polynomial extension. Part I, arXiv:1209.4110v1 [math.NT], Sep 18, 2012.

MAPLE

A216912 := n -> denom(add(binomial(2*n+r, 2*r)*bernoulli(r)/(2*n+r), r=0..2*n))/4;

seq(A216912(i), i=1..38); # Peter Luschny, Sep 20 2012

MATHEMATICA

a[n_] := Denominator[Sum[Binomial[2n+r, 2r]*(BernoulliB[r]/(2n+r)), {r, 0, 2n}]]/4;

Array[a, 38] (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)

PROG

(PARI) a(n) = denominator(sum(k=0, 2*n, binomial(2*n+k, 2*k)*bernfrac(k)/(2*n+k)))/4; \\ Michel Marcus, Jul 14 2018

CROSSREFS

Cf. A216922, A216923.

Sequence in context: A309454 A267903 A280039 * A175671 A222741 A069257

Adjacent sequences:  A216909 A216910 A216911 * A216913 A216914 A216915

KEYWORD

nonn

AUTHOR

Jonathan Vos Post, Sep 20 2012

EXTENSIONS

a(16)-a(38) from Peter Luschny, Sep 20 2012

STATUS

approved

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Last modified December 6 06:34 EST 2019. Contains 329784 sequences. (Running on oeis4.)