|
%I #11 Jun 26 2019 03:41:03
%S 1,0,4,24,24,560,2080,11088,74864,343536,2050344,11676280,61903776,
%T 363737712,2022013760,11335886864,65187410400,365627715968,
%U 2085523894756,11894205734280,67517852274384,386394626371680,2205027379874400,12602057718873040,72195482578935488,413235574714857360
%N a(n) = [x^n] (theta_3(x^(1/2))^n + theta_4(x^(1/2))^n)/2, where theta_3() and theta_4() are the Jacobi theta functions.
%H Seiichi Manyama, <a href="/A303333/b303333.txt">Table of n, a(n) for n = 0..1000</a>
%H J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1007/978-1-4757-2016-7">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, p. 118.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F a(n) = A297331(n,n).
%F a(n) ~ c * d^n / sqrt(n), where d = 5.84456473064455581274428417... and c = 0.14104739588693592503498... - _Vaclav Kotesovec_, Jun 26 2019
%t Table[SeriesCoefficient[(EllipticTheta[3, 0, x^(1/2)]^n + EllipticTheta[4, 0, x^(1/2)]^n)/2, {x, 0, n}], {n, 0, 25}]
%t Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n, {x, 0, 2 n}], {n, 0, 25}]
%t Table[SeriesCoefficient[EllipticTheta[3, 0, Sqrt[x]]^n, {x, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Jun 26 2019 *)
%Y Main diagonal of A297331.
%Y Cf. A066535.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 21 2018
|
