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A181131 Denominator of Integral_{x=0..+oo} Polylog(-n, -x)^2 for n > 0, with a(0) = 1.

%I A181131 #49 Mar 16 2020 13:04:30
%S A181131 1,3,15,105,105,231,15015,2145,36465,969969,4849845,10140585,10140585,
%T A181131 22287,3231615,7713865005,7713865005,90751353,218257003965,1641030105,
%U A181131 67282234305,368217318651,1841086593255,3762220429695,63957747304815,1546231253523
%N A181131 Denominator of Integral_{x=0..+oo} Polylog(-n, -x)^2 for n > 0, with a(0) = 1.
%C A181131 These are the denominators of the Bernoulli median numbers (see A212196). - _Peter Luschny_, May 04 2012
%H A181131 Vincenzo Librandi, <a href="/A181131/b181131.txt">Table of n, a(n) for n = 0..200</a>
%F A181131 a(n) = denominator((-1)^n/Pi^(2*n)*integral((log(t/(1-t))*log(1-1/t))^n dt,t=0,1)). - [_Gerry Martens_, May 25 2011]
%F A181131 a(n) = Denominator(Sum_{k=0..n} C(n,k)*Bern(n+k)). - _Vladimir Kruchinin_, Apr 06 2015
%p A181131 seq(denom(add(binomial(n,k)*bernoulli(n+k),k=0..n)), n=0..100); # _Robert Israel_, Jun 02 2015
%t A181131 Table[Denominator[Integrate[PolyLog[-n, -x]^2, {x, 0, Infinity}]], {n, 1, 18}]
%t A181131 max = 25; t[0] = Table[BernoulliB[n], {n, 0, 2*max}]; t[n_] := Differences[t[0], n]; a[n_] := t[n][[n + 1]] // Denominator; Table[a[n], {n, 0, max}] (* _Jean-François Alcover_, Jul 25 2013, after _Peter Luschny_ *)
%o A181131 (Sage) # uses[BernoulliMedian_list from A212196]
%o A181131 def A181131_list(n):
%o A181131     return [denominator(q) for q in BernoulliMedian_list(n)]
%o A181131 # _Peter Luschny_, May 04 2012
%o A181131 (PARI) a(n)=denominator(-subst(intformal(polylog(-n,-x)^2),'x,0)) \\ _Charles R Greathouse IV_, Jul 21 2014
%Y A181131 Cf. A181130, A212196.
%K A181131 nonn,frac
%O A181131 0,2
%A A181131 _Vladimir Reshetnikov_, Jan 23 2011
%E A181131 Offset set to 0, a(0) and a(19)..a(25) added by _Peter Luschny_, May 04 2012