A298420 Expansion of f(x, x) * f(x, x^2) * f(x^2, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

1, 3, 4, 5, 6, 8, 9, 6, 8, 8, 9, 12, 8, 12, 8, 13, 18, 8, 16, 12, 16, 13, 6, 16, 8, 18, 20, 16, 15, 12, 24, 18, 16, 16, 12, 20, 17, 18, 16, 16, 30, 24, 16, 12, 16, 21, 24, 16, 16, 18, 20, 32, 18, 28, 24, 27, 20, 16, 24, 12, 32, 30, 24, 16, 18, 32, 25, 18, 32

Offset

0,2

Comments

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

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

Formula

Expansion of phi(-x^2) * phi(-x^3) * phi(-x^6) / chi(-x)^3 in powers of x where phi(), chi() are Ramanujan theta functions.

Expansion of q^(-1/4) * eta(q^2)^5 * eta(q^3)^2 * eta(q^6) / (eta(q)^3 * eta(q^4) * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [3, -2, 1, -1, 3, -5, 3, -1, 1, -2, 3, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 48^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A298421.

Example

G.f. = 1 + 3*x + 4*x^2 + 5*x^3 + 6*x^4 + 8*x^5 + 9*x^6 + 6*x^7 + 8*x^8 + ...
G.f. = q + 3*q^9 + 4*q^17 + 5*q^25 + 6*q^33 + 8*q^41 + 9*q^49 + 6*q^57 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -q] QPochhammer[ -q, q]^2 QPochhammer[ q^3]^2 QPochhammer[ q^6, q^12], {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x, x]^3, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^6 + A) / (eta(x + A)^3 * eta(x^4 + A) * eta(x^12 + A)), n))};

Crossrefs

Cf. A298421.

Sequence in context: A327953 A352025 A112768 * A197354 A089399 A003619

Adjacent sequences: A298417 A298418 A298419 * A298421 A298422 A298423

Keyword

nonn

Author

Michael Somos, Jan 18 2018

Status

approved