A294592 a(n) = [x^n] (theta_3(x)/theta_4(x))^n, where theta_() is the Jacobi theta function.

1, 4, 32, 304, 3072, 32024, 340352, 3666016, 39878656, 437091892, 4819567552, 53401892240, 594093969408, 6631726263608, 74242911364864, 833237193123104, 9371924860764160, 105614054423502408, 1192210691317862048, 13478559927485340144, 152589996020498655232, 1729590806617202662528

Offset: 0

Author

Ilya Gutkovskiy, Nov 03 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..500
• Eric Weisstein's MathWorld, Jacobi Theta Functions
• Wikipedia, Theta function

Crossrefs

Cf. A007096, A014969, A014970, A014972, A066535, A080054, A103261, A270919, A289522, A291697.

Programs

Maple

S:= series((JacobiTheta3(0,x)/JacobiTheta4(0,x))^n,x,51):
seq(coeff(S,x,n),n=0..50); # Robert Israel, Nov 03 2017

Mathematica

Table[SeriesCoefficient[(EllipticTheta[3, 0, x]/EllipticTheta[4, 0, x])^n, {x, 0, n}], {n, 0, 21}]
Table[SeriesCoefficient[Product[((1 + x^(2 k + 1))/(1 - x^(2 k + 1)))^(2 n), {k, 0, n}], {x, 0, n}], {n, 0, 21}]
Table[SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^(2 n), {x, 0, n}], {n, 0, 21}]
(* Calculation of constant d: *) 1/r /. FindRoot[{s == EllipticTheta[3, 0, r*s]/EllipticTheta[4, 0, r*s], EllipticTheta[4, 0, r*s] + r*s*Derivative[0, 0, 1][EllipticTheta][4, 0, r*s] == r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s]}, {r, 1/10}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jan 17 2024 *)

Formula

a(n) = [x^n] Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^(2*n).

From Vaclav Kotesovec, Nov 05 2017: (Start)

a(n) ~ c * d^n / sqrt(n), where

d = 11.61255065799699699891360038489317237925475956178123836149123386457... and

c = 0.34456510029264878768512693687607064416428117641473856418257649837... (End)