login
A282213
Coefficients in q-expansion of (E_2^3*E_4 - 3*E_2^2*E_6 + 3*E_2*E_4^2 - E_4*E_6)/3456, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.
5
0, 1, 72, 756, 4672, 15750, 54432, 117992, 299520, 551853, 1134000, 1772892, 3532032, 4829006, 8495424, 11907000, 19173376, 24142482, 39733416, 47052740, 73584000, 89201952, 127648224, 148048056, 226437120, 246109375, 347688432, 402320520, 551258624, 594847710
OFFSET
0,3
COMMENTS
Multiplicative because A001158 is. - Andrew Howroyd, Jul 25 2018
LINKS
FORMULA
G.f.: phi_{6, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A282586(n) - 3*A282595(n) + 3*A282101(n) - A013974(n))/3456. - Seiichi Manyama, Feb 19 2017
a(n) = n^3*A001158(n) for n > 0. - Seiichi Manyama, Feb 19 2017
Sum_{k=1..n} a(k) ~ zeta(4) * n^7 / 7. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 31 2023: (Start)
Multiplicative with a(p^e) = p^(3*e) * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-3)*zeta(s-6). (End)
EXAMPLE
a(6) = 1^6*6^3 + 2^6*3^3 + 3^6*2^3 + 6^6*1^3 = 54432.
MATHEMATICA
terms = 30;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E2[x]^3*E4[x] - 3 E2[x]^2*E6[x] + 3 E2[x] E4[x]^2 - E4[x] E6[x])/3456 + O[x]^terms // CoefficientList[#, x]&
(* or: *)
Table[n^3*DivisorSigma[3, n], {n, 0, terms-1}] (* Jean-François Alcover, Feb 27 2018 *)
PROG
(PARI) a(n) = if (n, n^3*sigma(n, 3), 0); \\ Michel Marcus, Feb 27 2018
CROSSREFS
Cf. A282211 (phi_{4, 3}), this sequence (phi_{6, 3}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282586 (E_2^3*E_4), A282595 (E_2^2*E_6), A282101 (E_2*E_4^2), A013974 (E_4*E_6 = E_10).
Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), A282099 (n^2*sigma_3(n)), this sequence (n^3*sigma_3(n))
Cf. A013662.
Sequence in context: A097205 A253917 A008448 * A304828 A268780 A086579
KEYWORD
nonn,easy,mult
AUTHOR
Seiichi Manyama, Feb 09 2017
STATUS
approved