A006953 a(n) = denominator of Bernoulli(2n)/(2n).

12, 120, 252, 240, 132, 32760, 12, 8160, 14364, 6600, 276, 65520, 12, 3480, 85932, 16320, 12, 69090840, 12, 541200, 75852, 2760, 564, 2227680, 132, 6360, 43092, 6960, 708, 3407203800, 12, 32640, 388332, 120, 9372, 10087262640

Offset: 1

Author

Simon Plouffe and N. J. A. Sloane

Keywords

• nonn
• frac
• easy
• nice

Comments

a(n) are alternately divisible by 12 and 120, a(n)/(12, 120, 12, 120, 12, 120, ...) = 1, 1, 21, 2, 11, 273, ... . - Paul Curtz, Sep 13 2011 and Michel Marcus, Jan 05 2013

A141590/(2 before a(n+1)) = 1/2 + 1/12 - 1/120 + 1/252 is an old semi-convergent series for Euler's constants A001620 ("2 before a" meaning that one term, namely 2, is inserted before the sequence). This series is discussed in details in reference [Blagouchine, 2016], Sect. 3 and Fig. 3. - Paul Curtz, Sep 13 2011, Michel Marcus, Jan 05 2013 and Iaroslav V. Blagouchine, Sep 16 2015

a(n) = A006863(n)/2. - Michel Marcus, Jan 05 2013

Examples

			Sequence Bernoulli(2n)/(2n) (n >= 1) begins 1/12, -1/120, 1/252, -1/240, 1/132, -691/32760, 1/12, -3617/8160, ... .
		

References

• 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.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• 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).
• Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.
• R. D. Carmichael, Notes on the simplex theory of numbers, Bull. Amer. Math. Soc. 15 (1909), 217-223.
• 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
• A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function
• J. Sondow and E. W. Weisstein, MathWorld: Harmonic Number
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Numerators are given by A001067.

Programs

GAP

List([1..40], n-> DenominatorRat(Bernoulli(2*n)/(2*n)) ); # G. C. Greubel, Sep 19 2019

Magma

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

Maple

A006953_list := proc(n) 1/(1-1/exp(z)); series(%,z,2*n+4);
seq(denom((-1)^i*(2*i+1)!*coeff(%,z,2*i+1)),i=0..n) end;
A006953_list(35); # Peter Luschny, Jul 12 2012

Mathematica

Table[Denominator[BernoulliB[2n]/(2n)],{n,40}] (* Harvey P. Dale, Jan 12 2022 *)

PARI

a(n) = denominator(bernfrac(2*n)/(2*n)); \\ Michel Marcus, Apr 21 2016

Sage

[denominator(bernoulli(2*n)/(2*n)) for n in (1..40)] # G. C. Greubel, Sep 19 2019

Formula

Zeta(1-2*n) = -Bernoulli(2*n)/(2*n).

G.f. for Bernoulli(2*n)/(2*n) = A001067(n)/A006953(n): (-1)^n/((2*Pi)^(2*n)*(2*n)) * Integral_{t=0..1} log(1-1/t)^(2*n) dt. - Gerry Martens, May 18 2011

E.g.f.: a(n) = denominator((2*n+1)!*[x^(2*n+1)](1/(1-1/exp(x)))). - Peter Luschny, Jul 12 2012

Extensions

Previous Mathematica program replaced by Harvey P. Dale, Jan 12 2022