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

0,3

Comments

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

Links

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

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)

Example

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

Mathematica

a[0]=0; a[n_]:=(n^2)*DivisorSigma[3, n]; Table[a[n], {n, 0, 32}] (* Indranil Ghosh, Feb 21 2017 *)

Prog

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

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)).

Sequence in context: A219888 A233363 A030165 * A247840 A017342 A341544

Adjacent sequences: A282096 A282097 A282098 * A282100 A282101 A282102

Keyword

nonn,easy,mult

Author

Seiichi Manyama, Feb 06 2017

Status

approved