Question

$\sum _{k=1}^nk^2$

solve

Answer 1

Explanation:

Define gn(x)=∑nk=0(nk)2(2kk)xkgn(x)=∑k=0n(nk)2(2kk)xk for n=0,1,2,…n=0,1,2,…. Those numbers gn=gn(1)gn=gn(1) are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving gn(x)gn(x). For example, for any prime p>5p>5 we have

∑k=1p−1gk(−1)k≡0(modp2)and∑k=1p−1gk(−1)k2≡0(modp).∑k=1p−1gk(−1)k≡0(modp2)and∑k=1p−1gk(−1)k2≡0(modp).

This is similar to Wolstenholme’s classical congruences

∑k=1p−11k≡0(modp2)and∑k=1p−11k2≡0