A281373 Coefficients in q-expansion of (E_2*E_4 - E_6)^2/(300*(E_6^2-E_4^3)), where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

0, 1, 60, 1680, 30280, 405678, 4369680, 39729200, 315045840, 2230260741, 14340456648, 84870112272, 467160257760, 2411818867430, 11759239565472, 54457051387536, 240692336520352, 1019498573990610, 4152992658207660, 16319887656747248, 62032458633713904, 228608370781579488

Offset

0,3

Comments

This is (up to a constant factor), the function phi defined in Cohn (2017) (see phi on page 114 of the Notices version).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)

Henry Cohn, A conceptual breakthrough in sphere packing, arXiv preprint arXiv:1611.01685 [math.MG], 2016.

Henry Cohn, A conceptual breakthrough in sphere packing, Notices Amer. Math. Soc., 64:2 (2017), pp. 102-115.

Formula

a(n) ~ exp(4*Pi*sqrt(n)) / (14400 * sqrt(2) * Pi^2 * n^(7/4)). - Vaclav Kotesovec, Jun 06 2018

Maple

with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
t1:=series((e2*e4-e6)^2/518400, q, M+1);
t2:=series((e4^3-e6^2)/1728, q, M+1);
t3:=series(t1/t2, q, M+1);
seriestolist(t3);

Mathematica

terms = 22;
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]*E4[x] - E6[x])^2/(300*(E6[x]^2 - E4[x]^3)) + O[x]^terms // CoefficientList[#, x]& // Abs (* Jean-François Alcover, Feb 27 2018 *)

Crossrefs

Cf. A006352, A004009, A013973, A145094, A281371 (the numerator), A000594 (the denominator), A319134, A319294.

Sequence in context: A269196 A054331 A160349 * A053528 A269104 A017776

Adjacent sequences: A281370 A281371 A281372 * A281374 A281375 A281376

Keyword

nonn

Author

N. J. A. Sloane, Feb 04 2017

Status

approved