A181130 - OEIS

%I #37 Mar 16 2020 13:09:22

%S 1,2,8,8,32,6112,3712,362624,71706112,3341113856,79665268736,

%T 1090547664896,38770843648,106053090598912,5507347586961932288,

%U 136847762542978039808,45309996254420664320,3447910579774800362340352

%N Numerator of Integral_{x=0..+oo} Polylog(-n, -x)^2.

%C (-1)^n*a(n) is the numerator on the main diagonal of the (truncated) array described in A168516. - _Paul Curtz_, Jun 20 2011

%C These are - up to signs -  the numerators of the Bernoulli median numbers (see A212196). - _Peter Luschny_, May 04 2012

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ComputationAndAsymptoticsOfBernoulliNumbers">The computation and asymptotics of the Bernoulli numbers</a>.

%F a(n) = numerator((-1)^n/Pi^(2*n)*integral((log(t/(1-t))*log(1-1/t))^n dt,t=0,1)). - [_Gerry Martens_, May 25 2011]

%p seq(numer((-1)^n*add(binomial(n,k)*bernoulli(n+k),k=0..n)), n=1..30); # _Robert Israel_, Jun 02 2015

%t Table[Numerator[Integrate[PolyLog[-n, -x]^2, {x, 0, Infinity}]], {n, 1, 18}]

%o (Sage) # uses[BernoulliMedian_list from A212196]

%o def A181130_list(n): return [q.numerator() for q in BernoulliMedian_list(n)]

%o # _Peter Luschny_, May 04 2012

%o (PARI) a(n)=(-1)^n*sum(k=0,n,binomial(n,k)*bernfrac(n+k)) \\ _Charles R Greathouse IV_, Jun 03 2015

%Y Cf. A181131 (denominator), A212196.

%K nonn,frac

%O 1,2

%A _Vladimir Reshetnikov_, Jan 23 2011