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

0,3

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

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

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 *)

Crossrefs

Main diagonal of A297331.

Cf. A066535.

Sequence in context: A169688 A222595 A103225 * A324517 A370493 A137980

Adjacent sequences: A303330 A303331 A303332 * A303334 A303335 A303336

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 21 2018

Status

approved