A319134 Expansion of -((25*E_4^4 - 49*E_6^2*E4) + 48*E_6*E_4^2*E_2 + (-49*E_4^3 + 25*E_6^2)*E_2^2)/(3657830400*delta^2) where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively and delta is A000594.

1, 86, 3750, 109672, 2419462, 43021728, 643548464, 8343640624, 95835049605, 991606081332, 9364586280842, 81571540591968, 661034448807902, 5019357866562208, 35927279225314344, 243657157464337888, 1572638456431119570, 9696997279843999470, 57313953586222481126, 325672739267123628976

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..5000

H. Cohn, A. Kumar, S. Miller, D. Radchenko, M. Viazovska, The sphere packing problem in dimension 24, Annals of Mathematics, 185 (3) (2017), 1017-1033.

Wikipedia, Sphere packing

Formula

a(n) ~ exp(4*Pi*sqrt(2*n)) / (132300 * 2^(1/4) * Pi^2 * n^(23/4)). - Vaclav Kotesovec, Sep 12 2018

Example

((25*E_4^4 - 49*E_6^2*E4) + 48*E_6*E_4^2*E_2 + (-49*E_4^3 + 25*E_6^2)*E_2^2)/(delta^2) =  - 3657830400*q - 314573414400*q^2 - 13716864000000*q^3 - 401161575628800*q^4 - ... .

Mathematica

nmax = 25; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); Rest[CoefficientList[Series[-((25*E4[x]^4 - 49*E6[x]^2*E4[x]) + 48*E6[x]*E4[x]^2*E2[x] + (-49*E4[x]^3 + 25*E6[x]^2)* E2[x]^2) / (3657830400 * x^2 * QPochhammer[x]^48), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 12 2018 *)

Crossrefs

Cf. A000594, A006352 (E_2), A004009 (E_4), A013973 (E_6), A082558, A281373,

About the numerator: A282012 (E_4^4), A282287 (E_6^2*E_4), A282596 (E_6*E_4^2*E_2), A008411 (E_4^3), A280869 (E_6^2), A281374 (E_2^2).

Sequence in context: A017802 A035738 A017749 * A266823 A262471 A223739

Adjacent sequences: A319131 A319132 A319133 * A319135 A319136 A319137

Keyword

nonn

Author

Seiichi Manyama, Sep 11 2018

Status

approved