A282050 Coefficients in q-expansion of (E_4^2 - E_2*E_6)/1008, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

0, 1, 66, 732, 4228, 15630, 48312, 117656, 270600, 533637, 1031580, 1771572, 3094896, 4826822, 7765296, 11441160, 17318416, 24137586, 35220042, 47045900, 66083640, 86124192, 116923752, 148035912, 198079200, 244218775, 318570252, 389021400, 497449568, 594823350

Offset

0,3

Comments

Multiplicative because A001160 is. - Andrew Howroyd, Jul 23 2018

Links

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

Formula

a(n) = A145095(n)/504 for n > 0.

G.f.: phi_{6, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

a(n) = (A008410(n) - A282096(n))/1008. - Seiichi Manyama, Feb 10 2017

If p is a prime, a(p) = p^6 + p = A131472(p). - Seiichi Manyama, Feb 10 2017

a(n) = n*A001160(n) for n > 0. - Seiichi Manyama, Feb 18 2017

Sum_{k=1..n} a(k) ~ zeta(6) * n^7 / 7. - Amiram Eldar, Sep 06 2023

From Amiram Eldar, Oct 30 2023: (Start)

Multiplicative with a(p^e) = p^e * (p^(5*e+5)-1)/(p^5-1).

Dirichlet g.f.: zeta(s-1)*zeta(s-6). (End)

Example

a(6) = 1^6*6^1 + 2^6*3^1 + 3^6*2^1 + 6^6*1^1 = 48312.

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}];
(E4[x]^2 - E2[x]*E6[x])/1008 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
Table[n * DivisorSigma[5, n], {n, 0, 30}] (* Amiram Eldar, Sep 06 2023 *)

Prog

(PARI) a(n) = if(n < 1, 0, n * sigma(n, 5)); \\ Andrew Howroyd, Jul 23 2018

Crossrefs

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), this sequence (phi_{6, 1}), A282060 (phi_{8, 1}).

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A145095 (-q*E'_6), A008410 (E_4^2 = E_8), A282096 (E_2*E_6).

Cf. A001160, A013664, A131472.

Sequence in context: A129218 A046409 A128952 * A270258 A223410 A258917

Adjacent sequences: A282047 A282048 A282049 * A282051 A282052 A282053

Keyword

nonn,easy,mult

Author

Seiichi Manyama, Feb 05 2017

Status

approved