I didn't know who or what Faulhabers were, either (by name). (So had already been to the Wikipedia article.) I learned today what I hadn't known: That all the Faulh. polynomials F_n(x) for n odd are themselves polynomials in (x^2 + x)/2. Interesting! Now I'm wondering about a related matter: Given any rational p/q in Q - Z in lowest terms, then for any integer N we can form the sum S_p/q(N) = 1 + 2^(p/q) + ... + N^(p/q). Obviously (?) we can't hope for S_p/q(x) to be a polynomial in x, but I suspect it is a polynomial in x^(1/q), or S_1/q(x). Which brings up the question: Suppose P(x) are polynomials in Z[x] or Q[x]. When does there exist a polynomial X(x) (in one of those places) with P(x) = X(Q(x)) ??? Is this easy to determine? Something tells me differentiation might help. —Dan Fred Lunnon wrote: ----- For the benefit of anybody else who had never heard of this chap either, see Conway & Guy "Book of Numbers" https://en.wikipedia.org/wiki/Faulhaber's_formula https://www.emis.de/journals/AMEN/2018/AMEN-170803.pdf https://www.sciencedirect.com/science/article/pii/S0022247X06008791 -----