A213023 Expansion of psi(x)^2 * psi(-x^3) / chi(-x^2) in powers of x where psi(), chi() are Ramanujan theta functions.

1, 2, 2, 3, 2, 2, 4, 4, 5, 3, 4, 5, 4, 6, 4, 4, 5, 7, 5, 3, 6, 8, 8, 8, 6, 3, 7, 6, 10, 6, 5, 10, 4, 8, 7, 8, 10, 6, 9, 8, 5, 10, 10, 11, 6, 9, 11, 6, 12, 9, 8, 8, 10, 9, 6, 6, 15, 12, 9, 9, 6, 13, 10, 13, 10, 7, 14, 12, 12, 8, 7, 13, 10, 16, 9, 10, 10, 12

Offset: 0

Author

Michael Somos, Jun 03 2012

Keywords

• nonn

Comments

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

Examples

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

Crossrefs

Cf. A180312.

Programs

Mathematica

QP := QPochhammer; a[n_]:=SeriesCoefficient[(QP[q^2]^3*QP[q^3]*QP[q^4] *QP[q^12])/(QP[q]^2*QP[q^6]), {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^2 + A)^3 * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A)^2 * eta(x^6 + A)), n))}

Formula

Expansion of q^(-17/24) * eta(q^2)^3 * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q)^2 * eta(q^6)) in powers of q.

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

a(n) = A180312(3*n + 1).