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A001067
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Numerator of Bernoulli(2*n)/(2*n).
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41
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1, -1, 1, -1, 1, -691, 1, -3617, 43867, -174611, 77683, -236364091, 657931, -3392780147, 1723168255201, -7709321041217, 151628697551, -26315271553053477373, 154210205991661, -261082718496449122051, 1520097643918070802691, -2530297234481911294093
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OFFSET
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1,6
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COMMENTS
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It was incorrectly claimed that a(n) is "also numerator of "modified Bernoulli number" b(2n) = Bernoulli(2*n)/(2*n*n!)"; actually, the numerators of these fractions and the numerators of "modified Bernoulli numbers" (see A057868 for details) differ from each other and from this sequence. - Andrey Zabolotskiy, Dec 03 2022
Absolute values give denominators of constant terms of Fourier series of meromorphic modular forms E_k/Delta, where E_k is the normalized k th Eisenstein series [cf. Gunning or Serre references] and Delta is the normalized unique weight-twelve cusp form for the full modular group (the generating function of Ramanujan's tau function.) - Barry Brent (barrybrent(AT)iphouse.com), Jun 01 2009
|a(n)| is a product of powers of irregular primes (A000928), with the exception of n = 1,2,3,4,5,7. - Peter Luschny, Jul 28 2009
Conjecture: If there is a prime p such that 2*n+1 < p and p divides a(n), then p^2 does not divide a(n). This conjecture is true for p < 12 million. - Seiichi Manyama, Jan 21 2017
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 259, (6.3.18) and (6.3.19); also p. 810.
L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
R. Kanigel, The Man Who Knew Infinity, pp. 91-92.
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285.
J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973, p. 93.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 259, (6.3.18) and (6.3.19).
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FORMULA
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Zeta(1-2*n) = - Bernoulli(2*n)/(2*n).
G.f.: numerators of coefficients of z^(2*n) in z/(exp(z)-1). - Benoit Cloitre, Jun 02 2003
For 2 <= k <= 1000 and k != 7, the 2-order of the full constant term of E_k/Delta = 3 + ord_2(k - 7). - Barry Brent (barrybrent(AT)iphouse.com), Jun 01 2009
G.f. for Bernoulli(2*n)/(2*n) = a(n)/A006953(n): (-1)^n/((2*Pi)^(2*n)*(2*n))*integral(log(1-1/t)^(2*n) dt,t=0,1). - Gerry Martens, May 18 2011
E.g.f.: a(n) = numerator((2*n+1)!*[x^(2*n+1)](1/(1-1/exp(x)))). - Peter Luschny, Jul 12 2012
|a(n)| = numerator of Integral_{r=0..1} HurwitzZeta(1-n, r)^2 dr. More general: |Bernoulli(2*n)| = binomial(2*n,n)*n^2*I(n) for n >= 1 where I(n) denotes the integral. - Peter Luschny, May 24 2015
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EXAMPLE
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The sequence Bernoulli(2*n)/(2*n) (n >= 1) begins 1/12, -1/120, 1/252, -1/240, 1/132, -691/32760, 1/12, -3617/8160, ...
The sequence of modified Bernoulli numbers begins 1/48, -1/5760, 1/362880, -1/19353600, 1/958003200, -691/31384184832000, ...
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MAPLE
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A001067_list := proc(n) 1/(1-1/exp(z)); series(%, z, 2*n+4);
seq(numer((2*i+1)!*coeff(%, z, 2*i+1)), i=0..n) end:
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MATHEMATICA
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PROG
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(PARI) {a(n) = if( n<1, 0, numerator( bernfrac(2*n) / (2*n)))}; /* Michael Somos, Feb 01 2004 */
(Sage)
@CachedFunction
def S(n, k) :
if k == 0 :
if n == 0 : return 1
else: return 0
return S(n, k-1) + S(n-1, n-k)
def BernoulliDivN(n) :
if n == 0 : return 1
return (-1)^n*S(2*n-1, 2*n-1)/(4^n-16^n)
[BernoulliDivN(n).numerator() for n in (1..22)]
(Sage) [numerator(bernoulli(2*n)/(2*n)) for n in (1..25)] # G. C. Greubel, Sep 19 2019
(Magma) [Numerator(Bernoulli(2*n)/(2*n)):n in [1..40]]; // Vincenzo Librandi, Sep 17 2015
(GAP) List([1..25], n-> NumeratorRat(Bernoulli(2*n)/(2*n))); # G. C. Greubel, Sep 19 2019
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CROSSREFS
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KEYWORD
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sign,frac,nice
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AUTHOR
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STATUS
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approved
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