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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.
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%I #45 Aug 27 2024 09:13:00

%S 0,1,24,108,448,750,2592,2744,7680,9477,18000,15972,48384,30758,65856,

%T 81000,126976,88434,227448,137180,336000,296352,383328,292008,829440,

%U 484375,738192,787320,1229312,731670,1944000,953312,2064384,1724976

%N 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.

%C Multiplicative because A000203 is. - _Andrew Howroyd_, Jul 25 2018

%H Seiichi Manyama, <a href="/A282211/b282211.txt">Table of n, a(n) for n = 0..1000</a>

%H Joerg Arndt, <a href="http://arxiv.org/abs/1202.6525">On computing the generalized Lambert series</a>, arXiv:1202.6525v3 [math.CA], (2012).

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

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

%F a(n) = n^3*A000203(n) for n > 0. - _Seiichi Manyama_, Feb 19 2017

%F 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

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

%F From _Amiram Eldar_, Oct 31 2023: (Start)

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

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

%F G.f.: A(q) = Sum_{n >= 1} n^4*q^n*(q^(2*n) + 4*q^n + 1)/(1 - q^n)^4. - _Mamuka Jibladze_, Aug 27 2024

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

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

%o (PARI) a(n) = if (n==0, 0, n^3*sigma(n)); \\ _Michel Marcus_, Feb 21 2017

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

%Y 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).

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

%K nonn,easy,mult

%O 0,3

%A _Seiichi Manyama_, Feb 09 2017