A282097 Coefficients in q-expansion of (3*E_2*E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

0, 1, 12, 36, 112, 150, 432, 392, 960, 1053, 1800, 1452, 4032, 2366, 4704, 5400, 7936, 5202, 12636, 7220, 16800, 14112, 17424, 12696, 34560, 19375, 28392, 29160, 43904, 25230, 64800, 30752, 64512, 52272, 62424, 58800, 117936, 52022, 86640, 85176, 144000, 70602

Offset: 0

Author

Seiichi Manyama, Feb 06 2017

Keywords

• nonn
• easy
• mult

Comments

Multiplicative because A000203 is. - Andrew Howroyd, Jul 25 2018

Examples

			a(6) = 1^3*6^2 + 2^3*3^2 + 3^3*2^2 + 6^3*1^2 = 432.
		

Links

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

Crossrefs

Cf. this sequence (phi_{3, 2}), A282099 (phi_{5, 2}).

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282018 (E_2^3), A282019 (E_2*E_4).

Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), this sequence (n^2*sigma(n)), A282211 (n^3*sigma(n)).

Cf. A222171.

Programs

Magma

[0] cat [n^2*DivisorSigma(1, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018

Mathematica

a[0]=0;a[n_]:=(n^2)*DivisorSigma[1,n];Table[a[n],{n,0,41}] (* Indranil Ghosh, Feb 21 2017 *)
terms = 42; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(3*Ei[2]*Ei[4] - 2*Ei[6] - Ei[2]^3)/1728 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

PARI

a(n) = if (n==0, 0, n^2*sigma(n)); \\ Michel Marcus, Feb 21 2017

Formula

a(n) = (3*A282019(n) - 2*A013973(n) - A282018(n))/1728.

G.f.: phi_{3, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

a(n) = n^2*A000203(n) for n > 0. - Seiichi Manyama, Feb 19 2017

G.f.: Sum_{k>=1} k^3*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, May 02 2018

From Amiram Eldar, Oct 30 2023: (Start)

Multiplicative with a(p^e) = p^(2*e) * (p^(e+1)-1)/(p-1).

Dirichlet g.f.: zeta(s-2)*zeta(s-3).

Sum_{k=1..n} a(k) ~ (Pi^2/24) * n^4. (End)

From Peter Bala, Jan 22 2024: (Start)

a(n) = Sum_{1 <= i, j, k <= n} sigma_2( gcd(i, j, k, n) ).

a(n) = Sum_{1 <= i, j <= n} sigma_3( gcd(i, j, n) ).

a(n) = Sum_{d divides n} sigma_2(d) * J_3(n/d) = Sum_{d divides n} sigma_3(d) * J_2(n/d), where the Jordan totient functions J_2(n) = A007434(n) and J_3(n) = A059376(n). (End)