A258593 Expansion of (phi(x^2) * psi(x^2) / phi(-x)^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

1, 8, 46, 208, 805, 2776, 8742, 25584, 70450, 184232, 460832, 1108848, 2578295, 5814992, 12760598, 27317056, 57174768, 117223008, 235818894, 466154416, 906606234, 1736736024, 3280271526, 6114139616, 11255369609, 20478505104, 36849912318, 65619691088

Offset: 0

Author

Michael Somos, Nov 06 2015

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 8*x + 46*x^2 + 208*x^3 + 805*x^4 + 2776*x^5 + 8742*x^6 + ...
G.f. = q + 8*q^3 + 46*q^5 + 208*q^7 + 805*q^9 + 2776*q^11 + 8742*q^13 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A260186.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) (EllipticTheta[ 3, 0, x^2] EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0 ,x]^2)^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^2]^2 QPochhammer[ -x^2, x^2] / EllipticTheta[ 4, 0, x]^2)^2, {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^7 / (eta(x + A)^4 * eta(x^2 + A) * eta(x^8 + A)^2))^2, n))};

Formula

Expansion of (f(x^2)^2 / (chi(-x^2) * phi(-x)^2))^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

Expansion of q^(-1/2) * (eta(q^4)^7 / (eta(q)^4 * eta(q^2) * eta(q^8)^2))^2 in powers of q.

Euler transform of period 8 sequence [ 8, 10, 8, -4, 8, 10, 8, 0, ...].

-4 * a(n) = A260186(2*n + 1).

a(n) ~ exp(2*Pi*sqrt(n)) / (256*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017