A075180 - OEIS

A075180

Denominators from e.g.f. 1/(1-exp(-x)) - 1/x.

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

0,1

COMMENTS

Denominators of -zeta(-n), n >= 0, where zeta is Riemann's zeta function.

Numerators are +1, A060054(n+1), n >= 1.

REFERENCES

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.

LINKS

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.

FORMULA

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)

a(2*n-1) = A006953(n) for n >= 1. - Georg Fischer, Dec 01 2022

EXAMPLE

1/2, 1/12, 0, -1/120, 0, 1/252, 0, -1/240, 0, 1/132, 0, -691/32760, ...

MAPLE

a := n -> denom(bernoulli(n+1, 1)/(n+1)); # Peter Luschny, Apr 22 2009

MATHEMATICA

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 *)

PROG

(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 (%))