A298932 Expansion of f(-x^3)^3 * phi(-x^12) / (f(-x) * chi(-x^4)) in powers of x where phi(), chi(), f() are Ramanujan theta functions.

1, 1, 2, 0, 3, 2, 4, 0, 4, 4, 6, 0, 5, 3, 6, 0, 6, 4, 4, 0, 8, 4, 6, 0, 9, 6, 6, 0, 6, 6, 12, 0, 8, 4, 12, 0, 8, 7, 8, 0, 9, 6, 8, 0, 12, 8, 6, 0, 8, 6, 14, 0, 12, 6, 12, 0, 8, 8, 12, 0, 13, 6, 12, 0, 18, 10, 8, 0, 8, 12, 12, 0, 16, 7, 14, 0, 12, 8, 12, 0, 16

Offset

0,3

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/2) * eta(q^3)^3 * eta(q^8) * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^24)) in powers of q.

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

a(4*n + 3) = 0. a(3*n + 2) = 2 * A213607(n). a(n) = A298931(3*n). a(2*n) = A298933(n).

Example

G.f. = 1 + x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^6 + 4*x^8 + 4*x^9 + ...
G.f. = q + q^3 + 2*q^5 + 3*q^9 + 2*q^11 + 4*q^13 + 4*q^17 + 4*q^19 + ...

Mathematica

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

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^24 + A)), n))};

Crossrefs

Cf. A213607, A298931, A298933.

Sequence in context: A338101 A338490 A213607 * A089196 A208435 A208457

Adjacent sequences: A298929 A298930 A298931 * A298933 A298934 A298935

Keyword

nonn

Author

Michael Somos, Jan 29 2018

Status

approved