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

1, 3, 15, 105, 105, 231, 15015, 2145, 36465, 969969, 4849845, 10140585, 10140585, 22287, 3231615, 7713865005, 7713865005, 90751353, 218257003965, 1641030105, 67282234305, 368217318651, 1841086593255, 3762220429695, 63957747304815, 1546231253523

Offset: 0

Author

Vladimir Reshetnikov, Jan 23 2011

Keywords

• nonn
• frac

Comments

These are the denominators of the Bernoulli median numbers (see A212196). - Peter Luschny, May 04 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A181130, A212196.

Programs

Maple

seq(denom(add(binomial(n,k)*bernoulli(n+k),k=0..n)), n=0..100); # Robert Israel, Jun 02 2015

Mathematica

Table[Denominator[Integrate[PolyLog[-n, -x]^2, {x, 0, Infinity}]], {n, 1, 18}]
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 *)

PARI

a(n)=denominator(-subst(intformal(polylog(-n,-x)^2),'x,0)) \\ Charles R Greathouse IV, Jul 21 2014

Sage

# uses[BernoulliMedian_list from A212196]
def A181131_list(n):
    return [denominator(q) for q in BernoulliMedian_list(n)]
# Peter Luschny, May 04 2012

Formula

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]

a(n) = Denominator(Sum_{k=0..n} C(n,k)*Bern(n+k)). - Vladimir Kruchinin, Apr 06 2015

Extensions

Offset set to 0, a(0) and a(19)..a(25) added by Peter Luschny, May 04 2012