Congruences involving S_k^(2)(−1/2), S_k^(2)(−4), and S_k^(2)(9)

Prove the congruences and p-adic integrality assertions for the three related families of sums involving S_k^{(2)}(−1/2), S_k^{(2)}(−4), and S_k^{(2)}(9) specified in Conjecture 4.10.

Background

This conjecture groups several transformations and weighted sums whose predicted values are governed by quadratic forms involving 13.