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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. 3
0, 1, 60, 1680, 30280, 405678, 4369680, 39729200, 315045840, 2230260741, 14340456648, 84870112272, 467160257760, 2411818867430, 11759239565472, 54457051387536, 240692336520352, 1019498573990610, 4152992658207660, 16319887656747248, 62032458633713904, 228608370781579488 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)