A374930 Expansion of Sum_{1<=i<=j<=k} q^(i+j+k)/( (1-q^i)*(1-q^j)*(1-q^k) )^2.

1, 7, 27, 77, 181, 378, 707, 1254, 2052, 3290, 4928, 7371, 10381, 14756, 19818, 27158, 35139, 46683, 58806, 76146, 93555, 119092, 143222, 178983, 212408, 261261, 305046, 371931, 428156, 515592, 589385, 701442, 792720, 939918, 1050567, 1233387, 1374835, 1600143, 1766583, 2052898, 2247784

Offset: 3

Author

Seiichi Manyama, Jul 24 2024

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 3..10000
• Tewodros Amdeberhan, George E. Andrews and Roberto Tauraso, Extensions of MacMahon's sums of divisors, Research in the Mathematical Sciences, Vol. 11 (2024), Article number 8; arXiv preprint, arXiv:2309.03191 [math.CO], 2023.

Crossrefs

Cf. A374929, A374931.

Cf. A000203, A001158, A001160, A002128.

Programs

Mathematica

A374930[n_] := (31*DivisorSigma[5, n] - 70*(n + 1)*DivisorSigma[3, n] + (40*n^2 + 60*n + 9)*DivisorSigma[1, n])/1920;
Array[A374930, 50, 3] (* Paolo Xausa, Jul 24 2024 *)

PARI

a(n) = (31*sigma(n, 5)-70*(n+1)*sigma(n, 3)+(40*n^2+60*n+9)*sigma(n))/1920;

Python

from math import prod
from sympy import factorint
def A374930(n):
    f = factorint(n).items()
    return (31*prod((p**(5*(e+1))-1)//(p**5-1) for p,e in f)-70*(n+1)*prod((p**(3*(e+1))-1)//(p**3-1) for p,e in f) + (20*n*((n<<1)+3)+9)*prod((p**(e+1)-1)//(p-1) for p, e in f))//1920 # Chai Wah Wu, Jul 24 2024

Formula

a(n) = (31*sigma_5(n) - 70*(n+1)*sigma_3(n) + (40*n^2+60*n+9)*sigma(n))/1920.