A303333 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.
1, 0, 4, 24, 24, 560, 2080, 11088, 74864, 343536, 2050344, 11676280, 61903776, 363737712, 2022013760, 11335886864, 65187410400, 365627715968, 2085523894756, 11894205734280, 67517852274384, 386394626371680, 2205027379874400, 12602057718873040, 72195482578935488, 413235574714857360
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 118.
- Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Programs
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Mathematica
Table[SeriesCoefficient[(EllipticTheta[3, 0, x^(1/2)]^n + EllipticTheta[4, 0, x^(1/2)]^n)/2, {x, 0, n}], {n, 0, 25}] Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n, {x, 0, 2 n}], {n, 0, 25}] Table[SeriesCoefficient[EllipticTheta[3, 0, Sqrt[x]]^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jun 26 2019 *)
Formula
a(n) = A297331(n,n).
a(n) ~ c * d^n / sqrt(n), where d = 5.84456473064455581274428417... and c = 0.14104739588693592503498... - Vaclav Kotesovec, Jun 26 2019