From: Bernoulli numbers, convolution sums and congruences of coefficients for certain generating functions
f ( p ) : =
a
b
c
d
∑ k = 1 p − 1 σ 1 ∗ ( 2 m k) σ 1 ∗ ( 2 n (p−k))
0
3â‹… 2 m + n + 1
− 2 m + n + 1
∑ k = 1 p − 1 σ 1 ∗ ( 2 m k) σ 3 ∗ ( 2 n (p−k))
5â‹… 2 m + 3 n + 1
− 2 m + 3 n + 2
5â‹… 2 m + 3 n
− 2 m + 3 n
∑ r + s + t = p σ 1 ∗ ( 2 m r) σ 1 ∗ ( 2 n s) σ 1 ∗ ( 2 l t)
5⋅ 2 m + n + l − 1
−3⋅ 2 m + n + l
2 m + n + l + 1
− 2 m + n + l − 1