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

0,2

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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).

Example

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 + ...

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 *)

Prog

(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))}

Crossrefs

Cf. A180312.

Sequence in context: A339492 A069360 A175509 * A068050 A210967 A343534

Adjacent sequences: A213020 A213021 A213022 * A213024 A213025 A213026

Keyword

nonn

Author

Michael Somos, Jun 03 2012

Status

approved