A075180 - OEIS

%I #46 Dec 01 2022 17:29:33

%S 2,12,1,120,1,252,1,240,1,132,1,32760,1,12,1,8160,1,14364,1,6600,1,

%T 276,1,65520,1,12,1,3480,1,85932,1,16320,1,12,1,69090840,1,12,1,

%U 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

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

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

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

%D 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.

%H Antti Karttunen, <a href="/A075180/b075180.txt">Table of n, a(n) for n = 0..16384</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 807, combined eqs. 23.2.11,14 and 15.

%F 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).

%F 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.

%F From _Jianing Song_, Apr 05 2021: (Start)

%F 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.

%F 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.

%F a(2n-1) = A002445(n)*(2n)/A300711(n), n > 0. (End)

%F a(2*n-1) = A006953(n) for n >= 1. - _Georg Fischer_, Dec 01 2022

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

%p a := n -> denom(bernoulli(n+1,1)/(n+1)); # _Peter Luschny_, Apr 22 2009

%t 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 *)

%t Table[Denominator[Zeta[-n]], {n, 0, 49}] (* _Alonso del Arte_, Jan 13 2012 *)

%t CoefficientList[ Series[ EulerGamma - HarmonicNumber[n] + Log[n], {n, Infinity, 48}], 1/n] // Rest // Denominator (* _Jean-François Alcover_, Mar 28 2013 *)

%t 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 *)

%o (PARI)

%o x='x+O('x^66);

%o egf = 1/(1-exp(-x)) - 1/x;

%o v=Vec(serlaplace(egf));

%o vector(#v,n, denominator(v[n]))

%o /* _Joerg Arndt_, Mar 28 2013 */

%o (PARI) A075180(n) = denominator(bernfrac(n+1)/(n+1)); \\ _Antti Karttunen_, Dec 19 2018, after Maple-program.

%o (Haskell)

%o a075180 n = a075180_list !! n

%o a075180_list = map (denominator . sum) $ zipWith (zipWith (%))

%o    (zipWith (map . (*)) a000142_list a242179_tabf) a106831_tabf

%o -- _Reinhard Zumkeller_, Jul 04 2014

%Y Cf. A060054, A006232 with A006233.

%Y Cf. A006953, A006863, A079612, A242179, A106831, A000142, A300711.

%K nonn,easy,frac

%O 0,1

%A _Wolfdieter Lang_, Sep 06 2002

%E More terms from _Antti Karttunen_, Dec 19 2018