A282099 Coefficients in q-expansion of (E_2^2*E_4 - 2*E_2*E_6 + E_4^2)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

0, 1, 36, 252, 1168, 3150, 9072, 16856, 37440, 61317, 113400, 161172, 294336, 371462, 606816, 793800, 1198336, 1420146, 2207412, 2476460, 3679200, 4247712, 5802192, 6436872, 9434880, 9844375, 13372632, 14900760, 19687808, 20511990, 28576800, 28630112, 38347776

Offset: 0

Author

Seiichi Manyama, Feb 06 2017

Keywords

• nonn
• easy
• mult

Comments

Multiplicative because A001158 is. - Andrew Howroyd, Jul 25 2018

Examples

			a(6) = 1^5*6^2 + 2^5*3^2 + 3^5*2^2 + 6^5*1^2 = 9072.
		

Links

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

Crossrefs

Cf. A282097 (phi_{3, 2}), this sequence (phi_{5, 2}).

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282208 (E_2^2*E_4), A282096 (E_2*E_6), A008410 (E_8 = E_4^2).

Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), this sequence (n^2*sigma_3(n)), A282213 (n^3*sigma_3(n)).

Programs

Mathematica

a[0]=0;a[n_]:=(n^2)*DivisorSigma[3,n];Table[a[n],{n,0,32}] (* Indranil Ghosh, Feb 21 2017 *)
nmax = 40; CoefficientList[Series[Sum[k^5*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

PARI

a(n) = if (n==0, 0, n^2*sigma(n, 3)); \\ Michel Marcus, Feb 21 2017

Formula

G.f.: phi_{5, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

a(n) = (A282208(n) - 2*A282096(n) + A008410(n))/1728. - Seiichi Manyama, Feb 19 2017

a(n) = n^2*A001158(n) for n > 0. - Seiichi Manyama, Feb 19 2017

Sum_{k=1..n} a(k) ~ Pi^4 * n^6 / 540. - Vaclav Kotesovec, May 09 2022

From Amiram Eldar, Oct 30 2023: (Start)

Multiplicative with a(p^e) = p^(2*e) * (p^(3*e+3)-1)/(p^3-1).

Dirichlet g.f.: zeta(s-2)*zeta(s-5). (End)

G.f.: Sum_{k>=1} k^5*x^k*(1 + x^k)/(1 - x^k)^3. - Vaclav Kotesovec, Aug 02 2025