A006953 a(n) = denominator of Bernoulli(2n)/(2n). (Formerly M2039)

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

1,1

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

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.

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

Example

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

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 *)

Prog

(Magma) [Denominator(Bernoulli(2*n)/(2*n)):n in [1..40]]; // Vincenzo Librandi, Sep 17 2015
(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
(GAP) List([1..40], n-> DenominatorRat(Bernoulli(2*n)/(2*n)) ); # G. C. Greubel, Sep 19 2019

Crossrefs

Numerators are given by A001067.

Sequence in context: A276668 A076633 A110423 * A164877 A121032 A188251

Adjacent sequences: A006950 A006951 A006952 * A006954 A006955 A006956

Keyword

nonn,frac,easy,nice

Author

Simon Plouffe and N. J. A. Sloane

Extensions

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

Status

approved