Your case p=1 seems to be http://oeis.org/A078812, just binomial(n+k-1, 2*k-1). See also the references to http://oeis.org/A078812.
The case p=-1 seems to be a variant of http://oeis.org/A048594, more specifically
k!/n! S1(n,k) with S1 the unsigned Stirling numbers of the first kind.  
Forgive me for including a line of obtruse Mathematica 9.0 code:

Table[  Apply[Times,Map[#^p&,temp=IntegerPartitions[n,{k}],{2}],1] . 
(Apply[Multinomial,Length/@ Split[#]]&/@ temp),{n,8},{k,n}] /. p-> -1
is, as Brendan McKay pointed out, equivalent to
Table[SeriesCoefficient [PolyLog[-p, x]^k, {x, 0, n}], {n, 8}, {k, n}] /. p-> -1