A226861 Expansion of phi(x) * f(-x^3) in powers of x where phi(), f() are Ramanujan theta functions.

1, 2, 0, -1, 0, 0, -1, -4, 0, 2, -2, 0, -2, 0, 0, -1, 4, 0, 0, 0, 0, 1, 0, 0, 2, 4, 0, 0, -2, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, -2, 0, 0, -2, 0, 0, -3, 0, 0, 0, 0, 0, 2, -4, 0, -2, -2, 0, 2, 0, 0, 0, -4, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, 1, 4, 0, 0

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/8) * eta(q^2)^5 * eta(q^3) / (eta(q) * eta(q^4))^2 in powers of q.

Euler transform of period 12 sequence [2, -3, 1, -1, 2, -4, 2, -1, 1, -3, 2, -2, ...].

G.f.: (Sum_{k in Z} x^(k^2)) * Product_{k>0} (1 - x^(3*k)).

a(3*n + 2) = 0. a(3*n) = A226289(n).

Example

G.f. = 1 + 2*x - x^3 - x^6 - 4*x^7 + 2*x^9 - 2*x^10 - 2*x^12 - x^15 + 4*x^16 + ...
G.f. = q + 2*q^9 - q^25 - q^49 - 4*q^57 + 2*q^73 - 2*q^81 - 2*q^97 - q^121 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] QPochhammer[ q^3], {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) / (eta(x + A) * eta(x^4 + A))^2, n))};

Crossrefs

Cf. A226289.

Sequence in context: A286604 A366784 A217540 * A185643 A363051 A278515

Adjacent sequences: A226858 A226859 A226860 * A226862 A226863 A226864

Keyword

sign

Author

Michael Somos, Jun 20 2013

Status

approved