A261426 Expansion of f(-x^3)^3 * phi(x^6) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

1, 1, 2, 0, 2, 1, 4, 2, 5, 2, 6, 2, 6, 0, 4, 4, 7, 2, 4, 0, 6, 1, 8, 4, 4, 4, 10, 2, 8, 2, 12, 4, 8, 5, 6, 0, 14, 2, 8, 2, 11, 6, 6, 4, 8, 2, 8, 4, 8, 6, 14, 0, 6, 0, 12, 6, 15, 4, 14, 2, 14, 4, 8, 8, 12, 7, 14, 0, 12, 2, 16, 10, 8, 4, 10, 6, 14, 0, 16, 4, 16

Offset: 0

Author

Michael Somos, Aug 18 2015

Keywords

• nonn

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 + x + 2*x^2 + 2*x^4 + x^5 + 4*x^6 + 2*x^7 + 5*x^8 + 2*x^9 + ...
G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 4*q^19 + 2*q^22 + 5*q^25 + ...
		

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. A033687, A261394, A261426, A261444.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 EllipticTheta[ 3, 0, x^6] / QPochhammer[ x], {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^12 + A)^5 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2), n))};

Formula

Expansion of (1/3) * q^(-1/3) * c(q) * phi(q^6) in powers of q where phi() is a Ramanujan theta function and c() is a cubic AGM function. - Michael Somos, Sep 01 2015

Expansion of q^(-1/3) * eta(q^3)^3 * eta(q^12)^5 / (eta(q) * eta(q^6)^2 * eta(q^24)^2) in powers of q.

Euler transform of period 24 sequence [ 1, 1, -2, 1, 1, 0, 1, 1, -2, 1, 1, -5, 1, 1, -2, 1, 1, 0, 1, 1, -2, 1, 1, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (128/3)^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261426.

24 * a(n) = A261394(6*n + 2).

a(n) = A261444(2*n). Michael Somos, Sep 01 2015