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Table 9 Some convolution formulas for odd N

From: Bernoulli numbers, convolution sums and congruences of coefficients for certain generating functions

Convolution sum

Convolution formula

∑ k = 1 N σ 1 (2k−1) σ 5 (2N−(2k−1))

1 17 {32 σ 7 (N)−15b(N)}

∑ k = 1 N σ 3 (2k−1) σ 3 (2N−(2k−1))

1 17 {8 σ 7 (N)+9b(N)}

∑ k = 1 N σ 1 (2k−1) σ 7 (2N−(2k−1))

1 496 {2,176 σ 9 (N)+224d(2N)−57,344l(N)+7c(2N)−1,792c(N)}

∑ k = 1 N σ 3 (2k−1) σ 5 (2N−(2k−1))

1 496 {256 σ 9 (N)−32d(2N)+8,192l(N)−c(2N)+256c(N)}

∑ k = 1 N σ 5 (2k−1) σ 5 (2N−(2k−1))

1 691 {256 σ 11 (N)+435τ(N)}

∑ k = 1 N σ 1 (2k−1) σ 9 (2N−(2k−1))

1 691 {7,936 σ 11 (N)−7,245τ(N)}

∑ k = 1 N σ 3 (2k−1) σ 7 (2N−(2k−1))

1 691 {544 σ 11 (N)+147τ(N)}