A181130 Numerator of Integral_{x=0..+oo} Polylog(-n, -x)^2.

1, 2, 8, 8, 32, 6112, 3712, 362624, 71706112, 3341113856, 79665268736, 1090547664896, 38770843648, 106053090598912, 5507347586961932288, 136847762542978039808, 45309996254420664320, 3447910579774800362340352

Offset: 1

Author

Vladimir Reshetnikov, Jan 23 2011

Keywords

• nonn
• frac

Comments

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

These are - up to signs - the numerators of the Bernoulli median numbers (see A212196). - Peter Luschny, May 04 2012

Links

• Peter Luschny, The computation and asymptotics of the Bernoulli numbers.

Crossrefs

Cf. A181131 (denominator), A212196.

Programs

Maple

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

Mathematica

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

PARI

a(n)=(-1)^n*sum(k=0,n,binomial(n,k)*bernfrac(n+k)) \\ Charles R Greathouse IV, Jun 03 2015

Sage

# uses[BernoulliMedian_list from A212196]
def A181130_list(n): return [q.numerator() for q in BernoulliMedian_list(n)]
# Peter Luschny, May 04 2012

Formula

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]