A281371 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.

0, 0, 1, 36, 492, 3608, 18828, 74760, 250352, 717984, 1866558, 4365580, 9635472, 19639032, 38559416, 71222616, 128258496, 219619968, 370366101, 597550068, 955638824, 1471571136, 2253173892, 3335433368, 4932972864, 7064391840, 10133162774, 14128072488, 19743952032, 26864847352

Offset

0,4

Comments

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).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Henry Cohn, A conceptual breakthrough in sphere packing, arXiv preprint arXiv:1611.01685 [math.MG], 2016; also Notices Amer. Math. Soc., 64:2 (2017), pp. 102-115.

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);
seriestolist(t1);

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]*E4[x] - E6[x])^2/518400 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

Crossrefs

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

Sequence in context: A104671 A323549 A128986 * A090408 A268635 A233101

Adjacent sequences: A281368 A281369 A281370 * A281372 A281373 A281374

Keyword

nonn

Author

N. J. A. Sloane, Feb 04 2017

Status

approved