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

Author

N. J. A. Sloane, Feb 04 2017

Keywords

• nonn

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.

Crossrefs

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

Programs

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 *)

Formula

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