A281373 - OEIS

%I #26 Sep 16 2018 21:40:47

%S 0,1,60,1680,30280,405678,4369680,39729200,315045840,2230260741,

%T 14340456648,84870112272,467160257760,2411818867430,11759239565472,

%U 54457051387536,240692336520352,1019498573990610,4152992658207660,16319887656747248,62032458633713904,228608370781579488

%N 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.

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

%H Vaclav Kotesovec, <a href="/A281373/b281373.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vincenzo Librandi)

%H Henry Cohn, <a href="https://arxiv.org/abs/1611.01685">A conceptual breakthrough in sphere packing</a>,  arXiv preprint arXiv:1611.01685 [math.MG], 2016.

%H Henry Cohn, <a href="http://www.ams.org/journals/notices/201702/rnoti-p102.pdf">A conceptual breakthrough in sphere packing</a>, Notices Amer. Math. Soc., 64:2 (2017), pp. 102-115.

%F a(n) ~ exp(4*Pi*sqrt(n)) / (14400 * sqrt(2) * Pi^2 * n^(7/4)). - _Vaclav Kotesovec_, Jun 06 2018

%p with(numtheory); M:=100;

%p E := proc(k) local n, t1; global M;

%p t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);

%p series(t1, q, M+1); end;

%p e2:=E(2); e4:=E(4); e6:=E(6);

%p t1:=series((e2*e4-e6)^2/518400,q,M+1);

%p t2:=series((e4^3-e6^2)/1728,q,M+1);

%p t3:=series(t1/t2,q,M+1);

%p seriestolist(t3);

%t terms = 22;

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

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

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

%t (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 *)

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

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Feb 04 2017