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A280025
Expansion of phi_{7, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
2
0, 1, 144, 2268, 18688, 78750, 326592, 825944, 2396160, 4966677, 11340000, 19501812, 42384384, 62777078, 118935936, 178605000, 306774016, 410422194, 715201488, 894002060, 1471680000, 1873240992, 2808260928, 3405105288, 5434490880, 6152734375, 9039899232
OFFSET
0,3
COMMENTS
Multiplicative because A001158 is. - Andrew Howroyd, Jul 23 2018
LINKS
FORMULA
a(n) = n^4*A001158(n) for n > 0.
a(n) = (7*(A280024(n) - 4*A282780(n) + 6*A282752(n) - 4*A282102(n)) + 3*A008411(n) + 4*A280869(n))/41472.
Sum_{k=1..n} a(k) ~ c * n^8, where c = Pi^4/720 = 0.1352904... (= A152649 / 10). - Amiram Eldar, Dec 08 2022
From Amiram Eldar, Oct 31 2023: (Start)
Multiplicative with a(p^e) = p^(4*e) * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-4)*zeta(s-7). (End)
MATHEMATICA
Table[n^4 * DivisorSigma[3, n], {n, 0, 30}] (* Amiram Eldar, Oct 31 2023 *)
PROG
(PARI) a(n) = if(n < 1, 0, n^4 * sigma(n, 3)); \\ Andrew Howroyd, Jul 23 2018
CROSSREFS
Cf. A280022 (phi_{5, 4}), this sequence (phi_{7, 4}).
Cf. A280024 (E_2^4*E_4), A282780 (E_2^3*E_6), A282752 (E_2^2*E_4^2), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), A282099 (n^2*sigma_3(n)), A282213 (n^3*sigma_3(n)), this sequence (n^4*sigma_3(n)).
Cf. A152649.
Sequence in context: A066445 A008431 A250084 * A223687 A268625 A121344
KEYWORD
nonn,easy,mult
AUTHOR
Seiichi Manyama, Feb 22 2017
STATUS
approved