A266684 Expansion of f(-x) * f(-x^2)^4 / psi(x^3) in powers of x where psi(), f() are Ramanujan theta functions.

1, -1, -5, 3, 7, 4, 3, -18, -17, -1, 20, 36, -9, -14, -18, -12, 31, 16, -5, -54, -28, 6, 36, 72, 15, -21, -70, 3, 54, 28, -12, -90, -65, -12, 80, 72, 7, -38, -54, 42, 68, 40, 30, -126, -108, 4, 72, 144, -33, -43, -105, -48, 98, 52, 3, -144, -90, 18, 140, 180

Offset: 0

Author

Michael Somos, Jan 02 2016

Keywords

• sign
• look

Comments

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

Examples

			G.f. = 1 - x - 5*x^2 + 3*x^3 + 7*x^4 + 4*x^5 + 3*x^6 - 18*x^7 - 17*x^8 + ...
		

Links

• Alois P. Heinz, 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. A263021, A260301.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 2 q^(3/8) QPochhammer[ q] QPochhammer[ q^2]^4 / EllipticTheta[ 2, 0, q^(3/2)], {q, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A)^4 * eta(x^3 + A) / eta(x^6 + A)^2, n))};

Formula

Expansion of eta(q) * eta(q^2)^4 * eta(q^3) / eta(q^6)^2 in powers of q.

Euler transform of period 6 sequence [ -1, -5, -2, -5, -1, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 248832^(1/2) (t/I)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263021.

a(n) = A260301(3*n). a(3*n) = A260301(n).