A282211 Coefficients in q-expansion of (6*E_2^2*E_4 - 8*E_2*E_6 + 3*E_4^2 - E_2^4)/6912, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

0, 1, 24, 108, 448, 750, 2592, 2744, 7680, 9477, 18000, 15972, 48384, 30758, 65856, 81000, 126976, 88434, 227448, 137180, 336000, 296352, 383328, 292008, 829440, 484375, 738192, 787320, 1229312, 731670, 1944000, 953312, 2064384, 1724976

Offset

0,3

Comments

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

Links

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

Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Formula

G.f.: phi_{4, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

a(n) = (6*A282208(n) - 8*A282096(n) + 3*A008410(n) - A282210(n))/6912.

a(n) = n^3*A000203(n) for n > 0. - Seiichi Manyama, Feb 19 2017

G.f.: A(q) = Sum_{n >= 1} n^3*q^n*(q^(3*n) + 11*q^(2*n) + 11*q^n + 1)/(1 - q^n)^5. A faster converging series may be found by applying the operator x*d/dx once to equation 5 in Arndt, setting x = 1, and then applying the operator q*d/dq three times to the resulting equation. - Peter Bala, Jan 21 2021

Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^2/30 = 0.328986... . - Amiram Eldar, Dec 08 2022

From Amiram Eldar, Oct 31 2023: (Start)

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

Dirichlet g.f.: zeta(s-3)*zeta(s-4). (End)

Example

a(6) = 1^4*6^3 + 2^4*3^3 + 3^4*2^3 + 6^4*1^3 = 2592.

Mathematica

a[0]=0; a[n_]:=(n^3)*DivisorSigma[1, n]; Table[a[n], {n, 0, 33}] (* Indranil Ghosh, Feb 21 2017 *)

Prog

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

Crossrefs

Cf. this sequence (phi_{4, 3}), A282213 (phi_{6, 3}).

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282208 (E_2^2*E_4), A282096 (E_2*E_6), A008410 (E_4^2 = E_8), A282210 (E_2^4).

Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), A282097 (n^2*sigma(n)), this sequence (n^3*sigma(n)).

Sequence in context: A211591 A211585 A211599 * A103473 A162451 A307859

Adjacent sequences: A282208 A282209 A282210 * A282212 A282213 A282214

Keyword

nonn,easy,mult

Author

Seiichi Manyama, Feb 09 2017

Status

approved