Bern Police & Bernpolys

[Wouter Meeussen]

 got mildly interested in Sum[k^p,{k,0,n}] for a variable p and n.
Table[   Sum[k^p,{k,0,n}]   ,{p,7}] gives:

(n*(1 + n))/2
(n*(1 + n)*(1 + 2*n))/6
(n^2*(1 + n)^2)/4
(n*(1 + n)*(1 + 2*n)*(-1 + 3*n + 3*n^2))/30
(n^2*(1 + n)^2*(-1 + 2*n + 2*n^2))/12
(n*(1 + n)*(1 + 2*n)*(1 - 3*n + 6*n^3 + 3*n^4))/42
(n^2*(1 + n)^2*(2 - 4*n - n^2 + 6*n^3 + 3*n^4))/24

or, in other words,  -(-1)^p*(Zeta[-p] - Zeta[-p, -n]). 
In case n->Infinity this becomes Zeta[-p].
So I stumbled on

a=  Table[  -(-1)^p (-1)^(n+1) Pochhammer[p-n+1,n]Zeta[-p+n]/n! ,{p,10},{n,1,p}]
b=  Table[Drop[CoefficientList[BernoulliB[p,x],x],-1]/Range[p] ,{p,10}]
c=  Table[ Rest at Drop[CoefficientList[ Sum[k^p,{k,0,n}] ,n],-1] ,{p,10}] 

a==b and (b==c  except for the sign of the last coefficient in each row)

The coefficients of the Bernoulli polynomials (highest power dropped)
and multiplied by 2(p+1)!!  seems to be always integer :

Table[2(p+1)!! Drop[CoefficientList[BernoulliB[p,x],x],-1]/Range[p],{p,13}]

{-2}
{1, -3}
{0, 4, -8}
{-1, 0, 10, -15}
{0, -8, 0, 40, -48}
{5, 0, -35, 0, 105, -105}
{0, 64, 0, -224, 0, 448, -384}
{-63, 0, 420, 0, -882, 0, 1260, -945}
{0, -1152, 0, 3840, 0, -5376, 0, 5760, -3840}
{1575, 0, -10395, 0, 20790, 0, -20790, 0, 17325, -10395}
{0, 38400, 0, -126720, 0, 168960, 0, -126720, 0, 84480, -46080}
{-68409, 0, 450450, 0, -891891, 0, 849420, 0, -495495, 0, 270270, -135135}
{0, -2122752, 0, 6988800, 0, -9225216, 0, 6589440, 0, -3075072, 0, 1397760, -645120}

and I wonder if a smaller multiplicand could be found than 2(p+1)!!


wouter.