A374930 -id:A374930 - cp's OEIS frontend

A374929 Expansion of Sum_{1<=i<=j} q^(i+j)/( (1-q^i)*(1-q^j) )^2.

1, 5, 14, 29, 55, 86, 140, 191, 285, 355, 511, 595, 818, 938, 1240, 1356, 1810, 1905, 2471, 2640, 3297, 3410, 4415, 4399, 5495, 5690, 6930, 6895, 8718, 8440, 10416, 10480, 12438, 12220, 15295, 14421, 17435, 17402, 20595, 19670, 24272, 22715, 27433, 26939, 31110, 29716, 36735, 33714, 40180, 39210

Offset: 2

Seiichi Manyama, Jul 24 2024

nonn

Tewodros Amdeberhan, George E. Andrews and Roberto Tauraso, Extensions of MacMahon's sums of divisors, arXiv:2309.03191v1 [math.CO], Sep 06 2023.

Cf. A374930, A374931.

Cf. A000203, A001158, A002127.

A374929[n_] := (7*DivisorSigma[3, n] - (6*n + 1)*DivisorSigma[1, n])/24;
Array[A374929, 50, 2] (* Paolo Xausa, Jul 24 2024 *)

a(n) = (7*sigma(n, 3)-(6*n+1)*sigma(n))/24;

a(n) = (7*sigma_3(n) - (6*n+1)*sigma(n))/24.

A374931 Expansion of Sum_{1<=i<=j<=k<=l} q^(i+j+k+l)/( (1-q^i)(1-q^j)(1-q^k)*(1-q^l) )^2.

Original entry on oeis.org

1, 9, 44, 156, 450, 1121, 2508, 5139, 9867, 17831, 30888, 51117, 82212, 127458, 193809, 285702, 415701, 588797, 826803, 1133352, 1545986, 2062203, 2744811, 3579155, 4665375, 5964543, 7638983, 9600392, 12104136, 14992496, 18642996, 22787829, 28006341, 33830697, 41125050, 49183310, 59202486

Offset: 4

Seiichi Manyama, Jul 24 2024

nonn

Tewodros Amdeberhan, George E. Andrews and Roberto Tauraso, Extensions of MacMahon's sums of divisors, arXiv:2309.03191v1 [math.CO], Sep 06 2023.

Cf. A374929, A374930.

Cf. A365664.

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A374929 Expansion of Sum_{1<=i<=j} q^(i+j)/( (1-q^i)*(1-q^j) )^2.

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A374931 Expansion of Sum_{1<=i<=j<=k<=l} q^(i+j+k+l)/( (1-q^i)(1-q^j)(1-q^k)*(1-q^l) )^2.

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cp's OEIS Frontend

A374929 Expansion of Sum_{1<=i<=j} q^(i+j)/( (1-q^i)*(1-q^j) )^2.

Views

Author

Keywords

Links

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Programs

Mathematica

PARI

Formula

A374931 Expansion of Sum_{1<=i<=j<=k<=l} q^(i+j+k+l)/( (1-q^i)*(1-q^j)*(1-q^k)*(1-q^l) )^2.

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Author

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A374931 Expansion of Sum_{1<=i<=j<=k<=l} q^(i+j+k+l)/( (1-q^i)(1-q^j)(1-q^k)*(1-q^l) )^2.