A261251 Expansion of f(-x, -x) * f(-x^3, -x^15) / f(-x^6, -x^12)^2 in powers of x where f(,) is Ramanujan's general theta function.

1, -2, 0, -1, 4, 0, 2, -6, 0, -4, 8, 0, 7, -14, 0, -10, 24, 0, 14, -34, 0, -22, 48, 0, 33, -72, 0, -45, 104, 0, 62, -142, 0, -88, 192, 0, 122, -266, 0, -163, 364, 0, 216, -480, 0, -290, 632, 0, 386, -840, 0, -502, 1104, 0, 650, -1426, 0, -846, 1832, 0, 1093

Offset: 0

Author

Michael Somos, Aug 12 2015

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 2*x - x^3 + 4*x^4 + 2*x^6 - 6*x^7 - 4*x^9 + 8*x^10 + ...
G.f. = q - 2*q^3 - q^7 + 4*q^9 + 2*q^13 - 6*q^15 - 4*q^19 + 8*q^21 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A217786, A261240, A261252.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ x^(-3/4) EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^(9/2)] / (QPochhammer[ x^6] EllipticTheta[ 2, 0, x^(3/2)]), {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A) * eta(x^18 + A)^2 / (eta(x^2 + A) * eta(x^6 + A)^3 * eta(x^9 + A)), n))};

Formula

Expansion of phi(-x) * psi(x^9) / (f(-x^6) * psi(x^3)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Expansion of q^(-1/2) * eta(q)^2 * eta(q^3) * eta(q^18)^2 / (eta(q^2) * eta(q^6)^3 * eta(q^9)) in powers of q.

Euler transform of period 18 sequence [ -2, -1, -3, -1, -2, 1, -2, -1, -2, -1, -2, 1, -2, -1, -3, -1, -2, 0, ...].

a(3*n) = A261252(n). a(3*n + 1) = -2 * A217786(n). a(3*n + 2) = 0.

Convolution inverse of A261240.