A298931 Expansion of psi(x^4) * c(x^3) / (3*x) where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.

1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 2, 0, 3, 0, 0, 2, 2, 0, 4, 1, 0, 0, 2, 0, 4, 0, 0, 4, 1, 0, 6, 2, 0, 0, 2, 0, 5, 0, 0, 3, 3, 0, 6, 1, 0, 0, 4, 0, 6, 0, 0, 4, 5, 0, 4, 3, 0, 0, 2, 0, 8, 0, 0, 4, 3, 0, 6, 3, 0, 0, 4, 0, 9, 0, 0, 6, 4, 0, 6, 2, 0, 0, 4, 0, 6, 0, 0

Offset

0,7

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).

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 q^(-3/2) * eta(q^8)^2 * eta(q^9)^3 / (eta(q^3) * eta(q^4)) in powers of q.

Euler transform of a period 72 sequence.

A005872(2*n + 3) = 6*a(n). a(3*n) = A298932(n). a(3*n + 1) = A263452(n-1). a(3*n + 2) = a(4*n + 1) = 0.

Example

G.f. = q^3 + q^9 + q^11 + 2*q^15 + q^17 + 2*q^23 + 3*q^27 + 2*q^33 + ...
G.f. = 1 + x^3 + x^4 + 2*x^6 + x^7 + 2*x^10 + 3*x^12 + 2*x^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^8]^2 QPochhammer[ x^9]^3 / (QPochhammer[ x^3] QPochhammer[ x^4]), {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^8 + A)^2 * eta(x^9 + A)^3 / (eta(x^3 + A) * eta(x^4 + A)), n))};

Crossrefs

Cf. A005872, A263452, A298932.

Sequence in context: A113063 A123477 A035225 * A035219 A245716 A241425

Adjacent sequences: A298928 A298929 A298930 * A298932 A298933 A298934

Keyword

nonn

Author

Michael Somos, Jan 29 2018

Status

approved