A319306 - OEIS

%I #27 Sep 18 2018 07:48:21

%S 1,0,-232,0,86064,-1835008,23619232,-229638144,1841202076,

%T -12765888512,78856617456,-442924793856,2295931514240,-11106754756608,

%U 50583249259456,-218397947199488,899050944837546,-3545383150551040,13446464974112552,-49213617532305408

%N Expansion of (7 * theta_4(q)^20 * theta_2(q)^8 + 7 * theta_4(q)^24 * theta_2(q)^4 + 2 * theta_4(q)^28)/(2 * delta^2) in powers of q = exp(Pi i t), where delta is A000594.

%H Seiichi Manyama, <a href="/A319306/b319306.txt">Table of n, a(n) for n = -4..10000</a>

%H H. Cohn, A. Kumar, S. Miller, D. Radchenko, M. Viazovska, <a href="https://www.jstor.org/stable/26395748">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>

%e Let q = exp(Pi i t).

%e theta_2(q)^4 =     16*q          + 64*q^3 + ... .

%e theta_4(q)^4 = 1 -  8*q + 24*q^2 - 32*q^3 + ... .

%e delta = q^2 - 24*q^4 + 252*q^6 - 1472*q^8 + ... .

%e (7 * theta_4(q)^20 * theta_2(q)^8 + 7 * theta_4(q)^24 * theta_2(q)^4 + 2 * theta_4(q)^28)/delta^2

%e = 2*q^(-4) - 464*q^(-2) + 172128 - 3670016*q + 47238464*q^2 - 459276288*q^3 + ... .

%Y Cf. A000594, A007331, A008438 (theta_2(q)^4/(16*q)), A096727 (theta_4(q)^4), A319134, A319294, A319308 (theta_4(q)^20), A319309 (theta_4(q)^24), A319310 (theta_4(q)^28).

%K sign

%O -4,3

%A _Seiichi Manyama_, Sep 16 2018