A257399 Expansion of phi(x^3) * phi(-x^12) / chi(-x^4) in powers of x where phi(), chi() are Ramanujan theta functions.

1, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 2, 1, 0, 0, 4, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 4, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0

Offset: 0

Author

Michael Somos, Apr 21 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^3 + x^4 + 2*x^7 + x^8 + 2*x^11 + 2*x^12 + 2*x^16 + 3*x^20 + ...
G.f. = q + 2*q^19 + q^25 + 2*q^43 + q^49 + 2*q^67 + 2*q^73 + 2*q^97 + ...
		

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. A204531, A255317, A257398, A257400.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] EllipticTheta[ 3, 0, x^3] EllipticTheta[ 4, 0, x^12], {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^5 * eta(x^8 + A) / (eta(x^4 + A) * eta(x^3 + A)^2 * eta(x^24 + A)), n))};

Formula

Expansion of (phi(-x^24)^2 + 2 * x^3 * psi(-x^12)^2) / chi(-x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Expansion of q^(-1/6) * eta(q^6)^5 * eta(q^8) / (eta(q^4) * eta(q^3)^2 * eta(q^24)) in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A257400.

a(4*n + 1) = a(4*n + 2) = 0. a(4*n) = A257398(n). a(4*n + 3) = 2 * A255317(n).