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Coefficients of square root of theta series of D_4 (see A004011).
4

%I #19 Mar 31 2023 03:19:46

%S 1,12,-60,768,-11004,178200,-3093504,56265216,-1058194428,20410970124,

%T -401553531000,8026398749952,-162541338390528,3327702330562584,

%U -68761528402925568,1432192515405350400,-30037109244686774268,633790586271852392472,-13444940755220756447292,286577646482211381212928

%N Coefficients of square root of theta series of D_4 (see A004011).

%C Do these coefficients have a number-theoretic interpretation?

%H Seiichi Manyama, <a href="/A108096/b108096.txt">Table of n, a(n) for n = 0..735</a>

%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

%F a(n) ~ -(-1)^n * Gamma(1/4)^4 * exp(Pi*n) / (2^(7/2) * Pi^(7/2) * n^(3/2)). - _Vaclav Kotesovec_, Dec 10 2017

%F Convolution 4th power of this sequence gives A008658. - _Georg Fischer_, Mar 30 2023

%e More precisely, the theta series of D_4 begins 1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + 144*q^10 + 96*q^12 + ... and the square root of this is 1 + 12*q^2 - 60*q^4 + 768*q^6 - 11004*q^8 + 178200*q^10 - 3093504*q^12 + ...

%t CoefficientList[Series[Sqrt[EllipticTheta[3,0,x]^4 + EllipticTheta[2,0,x]^4], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Dec 10 2017 *)

%Y Cf. A004011, A008658, A108092.

%K sign

%O 0,2

%A _N. J. A. Sloane_ and _Michael Somos_, Jun 07 2005