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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. 3

%I #46 Sep 16 2018 16:41:31

%S 1,86,3750,109672,2419462,43021728,643548464,8343640624,95835049605,

%T 991606081332,9364586280842,81571540591968,661034448807902,

%U 5019357866562208,35927279225314344,243657157464337888,1572638456431119570,9696997279843999470,57313953586222481126,325672739267123628976

%N 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.

%H Seiichi Manyama, <a href="/A319134/b319134.txt">Table of n, a(n) for n = 1..5000</a>

%H H. Cohn, A. Kumar, S. Miller, D. Radchenko, M. Viazovska, <a href="https://www.jstor.org/stable/26395748?seq=1#page_scan_tab_contents">The sphere packing problem in dimension 24</a>, Annals of Mathematics, 185 (3) (2017), 1017-1033.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sphere_packing">Sphere packing</a>

%F a(n) ~ exp(4*Pi*sqrt(2*n)) / (132300 * 2^(1/4) * Pi^2 * n^(23/4)). - _Vaclav Kotesovec_, Sep 12 2018

%e ((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 - ... .

%t 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 *)

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

%Y 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).

%K nonn

%O 1,2

%A _Seiichi Manyama_, Sep 11 2018

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Last modified August 26 19:48 EDT 2024. Contains 375462 sequences. (Running on oeis4.)