A282060 Coefficients in q-expansion of E_4*(E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

0, 1, 258, 6564, 66052, 390630, 1693512, 5764808, 16909320, 43066413, 100782540, 214358892, 433565328, 815730734, 1487320464, 2564095320, 4328785936, 6975757458, 11111134554, 16983563060, 25801892760, 37840199712, 55304594136, 78310985304, 110992776480

Offset

0,3

Comments

Multiplicative because A013955 is. - Andrew Howroyd, Jul 25 2018

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: phi_{8, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

a(n) = (A282101(n) - A013974(n))/720. - Seiichi Manyama, Feb 10 2017

If p is a prime, a(p) = p^8 + p = A196288(p). - Seiichi Manyama, Feb 10 2017

a(n) = n*A013955(n) for n > 0. - Seiichi Manyama, Feb 18 2017

Sum_{k=1..n} a(k) ~ zeta(8) * n^9 / 9. - Amiram Eldar, Sep 06 2023

From Amiram Eldar, Oct 30 2023: (Start)

Multiplicative with a(p^e) = p^e * (p^(7*e+7)-1)/(p^7-1).

Dirichlet g.f.: zeta(s-1)*zeta(s-8). (End)

Example

a(6) = 1^8*6^1 + 2^8*3^1 + 3^8*2^1 + 6^8*1^1 = 1693512.

Mathematica

terms = 25;
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}];
E4[x]*(E2[x]*E4[x] - E6[x])/720 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
Table[n * DivisorSigma[7, n], {n, 0, 24}] (* Amiram Eldar, Sep 06 2023 *)

Prog

(PARI) a(n) = if(n < 1, 0, n*sigma(n, 7)) \\ Andrew Howroyd, Jul 25 2018

Crossrefs

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}, this sequence (phi_{8, 1}).

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282101 (E_2*E_4^2), A013974 (E_4*E_6 = E_10).

Cf. A013955, A013666, A196288.

Sequence in context: A158230 A209945 A233306 * A301546 A229330 A253636

Adjacent sequences: A282057 A282058 A282059 * A282061 A282062 A282063

Keyword

nonn,easy,mult

Author

Seiichi Manyama, Feb 05 2017

Status

approved