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A075180
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Denominators from e.g.f. 1/(1-exp(-x)) - 1/x.
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7
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2, 12, 1, 120, 1, 252, 1, 240, 1, 132, 1, 32760, 1, 12, 1, 8160, 1, 14364, 1, 6600, 1, 276, 1, 65520, 1, 12, 1, 3480, 1, 85932, 1, 16320, 1, 12, 1, 69090840, 1, 12, 1, 541200, 1, 75852, 1, 2760, 1, 564, 1, 2227680, 1, 132, 1, 6360, 1, 43092, 1, 6960, 1, 708, 1, 3407203800, 1, 12, 1, 32640, 1, 388332, 1, 120, 1, 9372, 1, 10087262640, 1, 12
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OFFSET
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0,1
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COMMENTS
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Denominators of -zeta(-n), n >= 0, where zeta is Riemann's zeta function.
Numerators are +1, A060054(n+1), n >= 1.
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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, Tenth Printing, 1972, p. 807, combined eqs. 23.2.11,14 and 15.
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 0..16384
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. 807, combined eqs. 23.2.11,14 and 15.
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FORMULA
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a(n) = denominator(-Zeta(-n)) = denominator(((-1)^(n+1))*B(n+1)/(n+1)), n >= 0, with Riemann's zeta function and the Bernoulli numbers B(n).
a(n) = denominators from e.g.f. (B(-x) - 1)/x, with B(x) = x/(exp(x) - 1), e.g.f. for Bernoulli numbers A027641(n)/A027642(n), n >= 0.
From Jianing Song, Apr 05 2021: (Start)
a(2n-1) = A006863(n)/2 for n > 0. By the comments in A006863, A006863(n) = A079612(2n) for n > 0. Hence a(n) = A079612(n+1)/2 all odd n. For all even n > 0, we have a(n) = 1, which is also equal to A079612(n+1)/2.
For odd n, a(n) is the product of p^(e+1) where p^e*(p-1) divides n+1 but p^(e+1)*(p-1) does not. For example, a(11) = 2^3 * 3^2 * 5^1 * 7^1 * 13^1 = 32760.
a(2n-1) = A002445(n)*(2n)/A300711(n), n > 0. (End)
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EXAMPLE
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1/2, 1/12, 0, -1/120, 0, 1/252, 0, -1/240, 0, 1/132, 0, -691/32760, ...
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MAPLE
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a := n -> denom(bernoulli(n+1, 1)/(n+1)); # Peter Luschny, Apr 22 2009
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MATHEMATICA
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a[m_] := Sum[(-2)^(-k-1) k! StirlingS2[m, k], k, 0, m}]/(2^(m+1)-1); Table[Denominator[a[i]], {i, 0, 20}] (* Peter Luschny, Apr 29 2009 *)
Table[Denominator[Zeta[-n]], {n, 0, 49}] (* Alonso del Arte, Jan 13 2012 *)
CoefficientList[ Series[ EulerGamma - HarmonicNumber[n] + Log[n], {n, Infinity, 48}], 1/n] // Rest // Denominator (* Jean-François Alcover, Mar 28 2013 *)
With[{nn=50}, Denominator[CoefficientList[Series[1/(1-Exp[-x])-1/x, {x, 0, nn}], x] Range[0, nn-1]!]] (* Harvey P. Dale, Apr 13 2016 *)
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PROG
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(PARI)
x='x+O('x^66);
egf = 1/(1-exp(-x)) - 1/x;
v=Vec(serlaplace(egf));
vector(#v, n, denominator(v[n]))
/* Joerg Arndt, Mar 28 2013 */
(PARI) A075180(n) = denominator(bernfrac(n+1)/(n+1)); \\ Antti Karttunen, Dec 19 2018, after Maple-program.
(Haskell)
a075180 n = a075180_list !! n
a075180_list = map (denominator . sum) $ zipWith (zipWith (%))
(zipWith (map . (*)) a000142_list a242179_tabf) a106831_tabf
-- Reinhard Zumkeller, Jul 04 2014
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CROSSREFS
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Cf. A060054, A006232 with A006233.
Cf. A006863, A079612, A242179, A106831, A000142, A300711.
Sequence in context: A107773 A221075 A211798 * A227830 A299521 A167164
Adjacent sequences: A075177 A075178 A075179 * A075181 A075182 A075183
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KEYWORD
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nonn,easy,frac,changed
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AUTHOR
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Wolfdieter Lang, Sep 06 2002
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EXTENSIONS
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More terms from Antti Karttunen, Dec 19 2018
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STATUS
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approved
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