A212907 Expansion of x^(-1/3) * psi(x^3) * c(x) / 3 in powers of x where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.

1, 1, 2, 1, 3, 3, 2, 2, 2, 5, 3, 3, 4, 4, 3, 4, 3, 5, 3, 6, 6, 3, 4, 5, 5, 7, 5, 4, 4, 4, 8, 3, 6, 7, 9, 7, 5, 4, 3, 9, 7, 4, 7, 5, 10, 5, 8, 8, 4, 7, 9, 6, 8, 7, 8, 10, 5, 6, 5, 9, 10, 7, 8, 6, 7, 10, 7, 12, 6, 10, 7, 5, 12, 6, 12, 14, 6, 6, 5, 11, 6, 8, 10

Offset: 0

Author

Michael Somos, Jun 16 2012

Keywords

• nonn

Comments

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

Examples

			1 + x + 2*x^2 + x^3 + 3*x^4 + 3*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 5*x^9 + ...
q^17 + q^41 + 2*q^65 + q^89 + 3*q^113 + 3*q^137 + 2*q^161 + 2*q^185 + ...
		

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

Programs

Mathematica

QP := QPochhammer; a[n_]:= SeriesCoefficient[(QP[q^3]*QP[q^6])^2/QP[q], {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jan 07 2018 *)

PARI

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

Formula

Expansion of q^(-17/24) * eta(q^3)^2 * eta(q^6)^2 / eta(q) in powers of q.

Euler transform of period 6 sequence [ 1, 1, -1, 1, 1, -3, ...].

G.f.: Product_{k>0} (1 - x^(3*k))^2 * (1 - x^(6*k))^2 / (1 - x^k).