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[#^q&,temp=IntegerPartitions[n,{k}],{2}],1] . (Apply[Multinomial,Length/@ Split[#]]&/@ temp),{n,8},{k,n}]/. q->-1

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

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.

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[#^q&,temp=IntegerPartitions[n,{k}],{2}],1] . (Apply[Multinomial,Length/@ Split[#]]&/@ temp),{n,8},{k,n}]/. q->-1