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
Examples
a(6) = 1^6*6^1 + 2^6*3^1 + 3^6*2^1 + 6^6*1^1 = 48312.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Crossrefs
Programs
-
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 *) nmax = 40; CoefficientList[Series[x*Sum[k^6*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 01 2025 *) -
PARI
a(n) = if(n < 1, 0, n * sigma(n, 5)); \\ Andrew Howroyd, Jul 23 2018
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}.
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)
G.f. Sum_{k>=1} k^6*x^(k-1)/(1 - x^k)^2. - Vaclav Kotesovec, Aug 01 2025
Comments