On an exam:
— Find the sum of the k-th degrees of the first N positive integers.
— That's easy. What's N?
— N is unknown, solve the problem in the general case.
— So how can I find this sum if N is unknown?
— We discussed it at the lectures. The sum
1k + 2k + 3k + … + Nk
for any k is a polynomial P(N) of degree k+1 with rational coefficients.
For example, 1 + … + N = N(N+1)/2. Given k,
find the coefficients of this polynomial.
Can you solve this problem?
Output coefficients of the polynomial
P(N) = Ak+1Nk+1 + AkNk + …
+A1N + A0
in the form of k+2 irreducible fractions.
A fraction has the form "a/b" or "−a/b",
where a and b are integers, b ≥ 1, a ≥ 0.
The coefficients must be given in the order of descending degrees
(from Ak+1 to A0).
It is not allowed to omit denominators of the fractions
or leave out zero coefficients.
Separate the fractions with a space.
Problem Author: Alexander Ipatov
Problem Source: Ural SU Contest. Petrozavodsk Winter Session, January 2006