A255505 -id:A255505 - cp's OEIS frontend

A001067 Numerator of Bernoulli(2n)/(2n).

1, -1, 1, -1, 1, -691, 1, -3617, 43867, -174611, 77683, -236364091, 657931, -3392780147, 1723168255201, -7709321041217, 151628697551, -26315271553053477373, 154210205991661, -261082718496449122051, 1520097643918070802691, -2530297234481911294093

Offset: 1

N. J. A. Sloane, Richard E. Borcherds (reb(AT)math.berkeley.edu)

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
Ramanujan incorrectly conjectured that the sequence contains only primes (and 1). - Jud McCranie. See A112548, A119766.
a(n) = A046968(n) if n < 574; a(574) = 37 * A046968(574). - Michael Somos, Feb 01 2004
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

			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, ...

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.

Seiichi Manyama, Table of n, a(n) for n = 1..314 (first 100 terms from T. D. Noe)
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).
D. Bar-Natan, T. T. Q. Le and D. P. Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, arXiv:math/0204311 [math.QA], 2002-2003; Geometry and Topology 7-1 (2003) 1-31.
G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3.
E. Z. Goren, Tables of values of Riemann zeta functions
Milan Janjic, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function
Eric Weisstein's World of Mathematics, Eisenstein Series.
Eric Weisstein's World of Mathematics, Bernoulli Number.
Wikipedia, Kummer-Vandiver conjecture
Index entries for sequences related to Bernoulli numbers

Similar to but different from A046968. See A090495, A090496.
Denominators given by A006953.
Cf. A000367, A006863, A033563, A046968.
Cf. A141590, A255505.

List([1..25], n-> NumeratorRat(Bernoulli(2*n)/(2*n)));  # G. C. Greubel, Sep 19 2019

[Numerator(Bernoulli(2*n)/(2*n)):n in [1..40]]; // Vincenzo Librandi, Sep 17 2015

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:
A001067_list(21); # Peter Luschny, Jul 12 2012

Table[ Numerator[ BernoulliB[2n]/(2n)], {n, 1, 22}] (* Robert G. Wilson v, Feb 03 2004 *)

{a(n) = if( n<1, 0, numerator( bernfrac(2*n) / (2*n)))}; /* Michael Somos, Feb 01 2004 */

@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)]
# Peter Luschny, Jul 08 2012

[numerator(bernoulli(2*n)/(2*n)) for n in (1..25)] # G. C. Greubel, Sep 19 2019

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

A141590 a(n) = numerator of Bernoulli(2n)/(2n + 1)!. Bisection of A120082.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, -691, 1, -3617, 43867, -174611, 77683, -236364091, 657931, -3392780147, 1723168255201, -7709321041217, 151628697551, -26315271553053477373, 154210205991661, -261082718496449122051, 1520097643918070802691, -2530297234481911294093

Offset: 0

Paul Curtz, Aug 20 2008

Numerators of the Taylor expansion coefficients of the Debye function D(1,x) at the even powers of x.

			Note that a(34) = -125235502160125163977598011460214000388469 but A255505(34) = -4633713579924631067171126424027918014373353.

Peter Luschny, Table of n, a(n) for n = 0..300
Kevin Acres and David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018.

Cf. A000367, A001067, A046968, A120082, A255505.

A141590:= func< n | Numerator(BernoulliNumber(2*n)/Factorial(2*n+1)) >;
[A141590(n): n in [0..35]]; // G. C. Greubel, Sep 16 2024

A141590 := proc(n) A120082(2*n) end:
seq(A141590(n), n=0..30) ; # R. J. Mathar, Sep 03 2009
seq(numer(bernoulli(2*n)/(2*n+1)!), n=0..34); # Peter Luschny, Dec 03 2022

Table[Numerator[BernoulliB[2*n]/(2*n+1)!], {n,0,35}] (* G. C. Greubel, Sep 16 2024 *)

def A141590(n): return numerator(bernoulli(2*n)/factorial(2*n+1))
[A141590(n) for n in range(36)] # G. C. Greubel, Sep 16 2024

a(n) = A120082(2*n).

Edited and extended by R. J. Mathar, Sep 03 2009

Edited by Peter Luschny, Dec 03 2022

A057868 Denominator of "modified Bernoulli number" b(2n) = Bernoulli(2n)/(4n(2n)!).

Original entry on oeis.org

48, 5760, 362880, 19353600, 958003200, 31384184832000, 2092278988800, 341459930972160000, 183927391818153984000, 32114306507931648000000, 620448401733239439360000, 81303558563123696133734400000, 9678995067038535254016000000, 2122022878497528469090467840000000

Offset: 1

Eric W. Weisstein

Note that Weisstein gives the formula b(n) = B(n)/(2*n*n!), and a(n) is the denominator of b(2*n). Numerators seem to be A141590 (not A001067 or A046968 or A255505). - Andrey Zabolotskiy, Dec 03 2022

			The sequence of modified Bernoulli numbers begins 1/48, -1/5760, 1/362880, -1/19353600, 1/958003200, -691/31384184832000, ...

D. Bar-Natan, T. T. Q. Le and D. P. Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geometry and Topology 7-1 (2003) 1-31.
Eric Weisstein's World of Mathematics, Modified Bernoulli Numbers.
Index entries for sequences related to Bernoulli numbers.

Numerators seem to be A141590.

Cf. A001067.

seq(denom(bernoulli(2*n)/((4*n)*(2*n)!)), n = 1..14); # Peter Luschny, Dec 03 2022

a[n_] := Denominator[ BernoulliB[2n] / (8n^2*(2n-1)!)];
Table[a[n], {n, 1, 12}] (* Jean-François Alcover, Jun 07 2012 *)

Name edited by Andrey Zabolotskiy, Dec 03 2022

A255506 Denominator of Bernoulli(2n)/(2n!).

Original entry on oeis.org

2, 12, 120, 504, 1440, 3168, 3931200, 8640, 41126400, 579156480, 2395008000, 1001548800, 2615348736000, 5748019200, 21670032384000, 7491404919398400, 21341245685760000, 251073478656000, 24574743184592240640000, 76828484468736000, 65834328341259878400000

Offset: 0

Jean-François Alcover, Feb 24 2015

			The sequence Bernoulli(2n)/(2n!) (n >= 0) begins 1/2, 1/12, -1/120, 1/504, -1/1440, 1/3168, -691/3931200, 1/8640, -3617/41126400, ...

MathOverflow, What can be said about a function given its asymptotic expansion?

Cf. A255505 (numerator).

[Denominator(Bernoulli(2*n)/(2*Factorial(n))): n in [0..25]]; // Vincenzo Librandi, Feb 24 2015

Table[Denominator[BernoulliB[2 n]/(2 n!)], {n, 0, 25}]

a(n) = denominator(bernfrac(2*n)/(2*n!)); \\ Michel Marcus, Feb 24 2015

[denominator(bernoulli(2*n)/(2*factorial(n))) for n in (0..25)] # Bruno Berselli, Feb 24 2015

cp's OEIS Frontend

A001067 Numerator of Bernoulli(2*n)/(2*n).

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A141590 a(n) = numerator of Bernoulli(2*n)/(2*n + 1)!. Bisection of A120082.

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A057868 Denominator of "modified Bernoulli number" b(2n) = Bernoulli(2*n)/(4*n*(2*n)!).

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A255506 Denominator of Bernoulli(2n)/(2n!).

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A001067 Numerator of Bernoulli(2n)/(2n).

A141590 a(n) = numerator of Bernoulli(2n)/(2n + 1)!. Bisection of A120082.

A057868 Denominator of "modified Bernoulli number" b(2n) = Bernoulli(2n)/(4n(2n)!).