A263528 Expansion of (psi(x) * psi(x^3) / f(-x^3)^2)^2 in powers of x where psi(), f() are Ramanujan theta functions.

1, 2, 1, 8, 14, 6, 38, 60, 23, 140, 208, 76, 439, 626, 221, 1232, 1704, 584, 3182, 4300, 1443, 7700, 10212, 3368, 17673, 23076, 7497, 38808, 50008, 16046, 82070, 104560, 33190, 167996, 211920, 66628, 334202, 417902, 130288, 648224, 804254, 248858, 1229148

Offset: 0

Author

Michael Somos, Oct 19 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*x + x^2 + 8*x^3 + 14*x^4 + 6*x^5 + 38*x^6 + 60*x^7 + 23*x^8 + ...
G.f. = q + 2*q^3 + q^5 + 8*q^7 + 14*q^9 + 6*q^11 + 38*q^13 + 60*q^15 + 23*x^17 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A262930.

Programs

Mathematica

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

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^6 + A)^2 / (eta(x + A) * eta(x^3 + A)^3))^2, n))};

Formula

Expansion of q^(-1/2) * (eta(q^2)^2 * eta(q^6)^2 / (eta(q) * eta(q^3)^3))^2 in powers of q.

Euler transform of period 6 sequence [ 2, -2, 8, -2, 2, 0, ...].

-2 * a(n) = A262930(2*n + 1).