assume that 1^k + 2^k + ... +n^k is a polynomial of degree k+1 in n
i.e. 1^k + 2^k +... +n^k = an^(k+1) +bn^k +cn^(k-1) + ...
what can you say about the coefficients of this polynomial? try to determine them as completely as possible
I need help with this! I can see that they will always add up to 1 (take n = 1) and obviously you could create a bunch of equations and solve them for specific n and k, but what else is there?
Let be the number of surjective functions from a set of m elements to a set of n elements. ( it's easy to see that where is the stirling number of the second kind)
Let and be sets such that . The number of functions from A to B is: . Now the number of functions with is .
But we have the following identity(see here ) :
Thus we get:
And that shows it's a polynomial of .
In fact from the formula we find that since