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A282213 Coefficients in q-expansion of (E_2^3*E_4 - 3*E_2^2*E_6 + 3*E_2*E_4^2 - E_4*E_6)/3456, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively. 5
0, 1, 72, 756, 4672, 15750, 54432, 117992, 299520, 551853, 1134000, 1772892, 3532032, 4829006, 8495424, 11907000, 19173376, 24142482, 39733416, 47052740, 73584000, 89201952, 127648224, 148048056, 226437120, 246109375, 347688432, 402320520, 551258624, 594847710 (list; graph; refs; listen; history; text; internal format)
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_{6, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

a(n) = (A282586(n) - 3*A282595(n) + 3*A282101(n) - A013974(n))/3456. - Seiichi Manyama, Feb 19 2017

a(n) = n^3*A001158(n) for n > 0. - Seiichi Manyama, Feb 19 2017

EXAMPLE

a(6) = 1^6*6^3 + 2^6*3^3 + 3^6*2^3 + 6^6*1^3 = 54432.

MATHEMATICA

terms = 30;

E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];

E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];

E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];

(E2[x]^3*E4[x] - 3 E2[x]^2*E6[x] + 3 E2[x] E4[x]^2 - E4[x] E6[x])/3456 + O[x]^terms // CoefficientList[#, x]&

(* or: *)

Table[n^3*DivisorSigma[3, n], {n, 0, terms-1}] (* Jean-Fran├žois Alcover, Feb 27 2018 *)

PROG

(PARI) a(n) = if (n, n^3*sigma(n, 3), 0); \\ Michel Marcus, Feb 27 2018

CROSSREFS

Cf. A282211 (phi_{4, 3}), this sequence (phi_{6, 3}).

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282586 (E_2^3*E_4), A282595 (E_2^2*E_6), A282101 (E_2*E_4^2), A013974 (E_4*E_6 = E_10).

Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), A282099 (n^2*sigma_3(n)), this sequence (n^3*sigma_3(n)).

Sequence in context: A097205 A253917 A008448 * A304828 A268780 A086579

Adjacent sequences:  A282210 A282211 A282212 * A282214 A282215 A282216

KEYWORD

nonn,mult

AUTHOR

Seiichi Manyama, Feb 09 2017

STATUS

approved

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Last modified March 30 19:49 EDT 2020. Contains 333127 sequences. (Running on oeis4.)