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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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,mult

AUTHOR

Seiichi Manyama, Feb 09 2017

STATUS

approved

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Last modified October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)