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A006953
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Denominator of Bernoulli(2*n)/(2*n).
(Formerly M2039)
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7
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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
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OFFSET
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1,1
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COMMENTS
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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 5 2013
A141590/(2 before a(n+1)) =1/2 + 1/12 -1/120 + 1/252 is an old Euler's formula for gamma A001620; "2 before a" meaning that one term (namely 2) is inserted before the sequence. - Paul Curtz, Sep 13 2011 and Michel Marcus, Jan 5 2013
a(n) = A006863(n)/2. - Michel Marcus, Jan 5 2013
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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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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).
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
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Riemann Zeta Function.
Eric Weisstein's World of Mathematics, Harmonic Number
Index entries for sequences related to Bernoulli numbers.
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FORMULA
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Zeta(1-2n) = - Bernoulli(2n)/(2n).
G.f. for Bernoulli(2*n)/(2*n) = A001067(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) = denominator((2*n+1)!*[x^(2*n+1)](1/(1-1/exp(x)))). - Peter Luschny, Jul 12 2012
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EXAMPLE
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Sequence Bernoulli(2n)/(2n) (n >= 1) begins 1/12, -1/120, 1/252, -1/240, 1/132, -691/32760, 1/12, -3617/8160, ...
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MAPLE
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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
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MATHEMATICA
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a[n_]:=Denominator[BernoulliB[2*n]/(2*n)]; [From Vladimir Joseph Stephan Orlovsky, Dec 13 2008]
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CROSSREFS
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Numerators given by A001067.
Sequence in context: A075366 A076633 A110423 * A164877 A121032 A188251
Adjacent sequences: A006950 A006951 A006952 * A006954 A006955 A006956
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KEYWORD
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nonn,frac,easy,nice,changed
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AUTHOR
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Simon Plouffe and N. J. A. Sloane.
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STATUS
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approved
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