A258099 Expansion of ( psi(x^3) * phi(-x^3) / (psi(x) * f(-x^2)) )^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

1, -2, 5, -12, 26, -50, 92, -168, 295, -496, 818, -1332, 2126, -3324, 5126, -7824, 11793, -17548, 25857, -37788, 54734, -78578, 111968, -158496, 222842, -311224, 432095, -596676, 819504, -1119624, 1522282, -2060448, 2776514, -3725294, 4978142, -6626988

Offset: 0

Author

Michael Somos, May 20 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 + 5*x^2 - 12*x^3 + 26*x^4 - 50*x^5 + 92*x^6 - 168*x^7 + ...
G.f. = q - 2*q^4 + 5*q^7 - 12*q^10 + 26*q^13 - 50*q^16 + 92*q^19 + ...
		

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. A132977, A258100.

Programs

Mathematica

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

PARI

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

Formula

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

Euler transform of period 6 sequence [ -2, 4, -4, 4, -2, 0, ...].

G.f.: Product_{k>0} (1 - x^k + x^(2*k))^2 * (1 + x^k + x^(2*k))^4 / (1 + x^k)^4.

a(n) = A258100(3*n + 1).

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018