sum of an exponential series (i know this is cliche, but it's urgent)

Okay, I'm brand new to this forum. I'm supposed to formulate a conjecture for the series (1^k)+(2^k)+(3^k)+(4^k)+...+(n^k)...

I've already found that the formula when k=1 is [n(n+1)/2]

when k=2 the sum of the series is (n/6)(n+1)(2n+1)

for k=3 it is [(n^2)/4](n+1)(n+1)

and for k=4, (n/30)(n+1)(2n+1)(3(n^2)+3n-1)

I don't know where to go from here. I've gone on to find the sums when k=5,6 but the question is based on the four sums I put above. Any ideas? Anyone done something like this before?