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

Author

Michael Somos, Jan 29 2018

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. = 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 + ...
		

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. A005872, A263452, A298932.

Programs

Mathematica

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

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))};

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.