A281371 - OEIS

%I #22 Jan 05 2023 11:05:13

%S 0,0,1,36,492,3608,18828,74760,250352,717984,1866558,4365580,9635472,

%T 19639032,38559416,71222616,128258496,219619968,370366101,597550068,

%U 955638824,1471571136,2253173892,3335433368,4932972864,7064391840,10133162774,14128072488,19743952032,26864847352

%N Coefficients in q-expansion of (E_2*E_4 - E_6)^2/518400, 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 numerator of the expression phi defined in Cohn (2017) (see phi on page 114 of the Notices version).

%H Seiichi Manyama, <a href="/A281371/b281371.txt">Table of n, a(n) for n = 0..1000</a>

%H Henry Cohn, <a href="http://arxiv.org/abs/1611.01685">A conceptual breakthrough in sphere packing</a>, arXiv preprint arXiv:1611.01685 [math.MG], 2016; also Notices Amer. Math. Soc., 64:2 (2017), pp. 102-115.

%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 seriestolist(t1);

%t terms = 30;

%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])^2/518400 + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 27 2018 *)

%Y Cf. A006352, A004009, A013973, A145094, A281372 (the square root).

%K nonn

%O 0,4

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