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A083688 Denominator of B(2n)*H(2n)/n where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.

%I A083688 #44 Jun 27 2025 09:21:39
%S A083688 4,144,360,33600,15120,34927200,2162160,172972800,1543782240,
%T A083688 10242872640,10346336,2338727174784,53542288800,4818805992000,
%U A083688 3228118134040800,1178332991611776000,78765574305600,12256711017694416000,2914326249307200,3205758874237920000,358462128664785600
%N A083688 Denominator of B(2n)*H(2n)/n where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
%C A083688 B(2n) is negative for even n, but this does not affect the denominator. - _M. F. Hasler_, Dec 24 2013
%H A083688 Seiichi Manyama, <a href="/A083688/b083688.txt">Table of n, a(n) for n = 1..500</a> (terms 1..50 from M. F. Hasler)
%H A083688 Ira Gessel, <a href="https://doi.org/10.1016/j.jnt.2003.08.010">On Miki's identity for Bernoulli numbers</a> J. Number Theory 110 (2005), no. 1, 75-82.
%F A083688 Miki's identity : B(n)*H(n)*(2/n) = sum(i=2, n-2, B(i)/i*B(n-i)/(n-i)*(1-C(n, i)))
%t A083688 Denominator[Table[(BernoulliB[2n]HarmonicNumber[2n])/(n (-1)^(n+1)),{n,20}]] (* _Harvey P. Dale_, Jun 25 2013 *)
%o A083688 (PARI) a(n)=denominator(bernfrac(2*n)*sum(k=1,2*n,1/k)/n)
%o A083688 (Python)
%o A083688 from sympy import bernoulli, harmonic
%o A083688 def a(n): return (bernoulli(2*n) * harmonic(2*n) / n).denominator
%o A083688 print([a(n) for n in range(1, 22)]) # _Indranil Ghosh_, Aug 04 2017
%Y A083688 Cf. A083687.
%K A083688 frac,nonn
%O A083688 1,1
%A A083688 _Benoit Cloitre_, Jun 15 2003