%I #24 Dec 03 2019 03:54:21

%S 1,5,7,761,671,4572347,1171733,518413759,32956355893,1949885751497,

%T 21495895979,63715389517501781,22630025105469577,36899945775958445129,

%U 517210776697519633301437,4518133367201930332907311663

%N Numerator of B(2n)*H(2n)/n*(-1)^(n+1) where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.

%H Seiichi Manyama, <a href="/A083687/b083687.txt">Table of n, a(n) for n = 1..260</a>

%H 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 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 Table[ BernoulliB[2n] * HarmonicNumber[2n] / n // Numerator // Abs, {n, 1, 16}] (* _Jean-François Alcover_, Mar 24 2015 *)

%o (PARI) a(n)=numerator((-1)^(n+1)*bernfrac(2*n)*sum(k=1,2*n,1/k)/n)

%o (Python)

%o from sympy import bernoulli, harmonic, numer

%o def a(n):

%o return numer(bernoulli(2 * n) * harmonic(2 * n) * (-1)**(n + 1) / n)

%o [a(n) for n in range(1, 31)] # _Indranil Ghosh_, Aug 04 2017

%Y Cf. A083688.

%K frac,nonn

%O 1,2

%A _Benoit Cloitre_, Jun 15 2003