A257873 Expansion of f(-x)^2 * chi(-x^4) * psi(x^6) in powers of x where psi(), chi(), f() are Ramanujan theta functions.

1, -2, -1, 2, 0, 4, 0, -4, -4, -2, 3, 4, 0, 4, 0, -8, 5, -6, 0, 6, 0, 4, 0, -4, -4, -8, -4, 10, 0, 8, 0, -4, 9, -6, -4, 6, 0, 8, 0, -8, -12, -12, 3, 6, 0, 12, 0, -12, 8, -6, 12, 8, 0, 8, 0, -12, -8, -10, -4, 6, 0, 12, 0, -8, 8, -10, -5, 16, 0, 8, 0, -12, -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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of phi(-x) * phi(-x^4) * f(-x^2, -x^10) in powers of x where phi(), f() are Ramanujan theta functions.

Expansion of q^(-2/3) * eta(q)^2 * eta(q^4) * eta(q^12)^2 / (eta(q^6) * eta(q^8)) in powers of q.

2 * a(n) = A127786(3*n + 2). a(8*n + 4) = a(8*n + 6) = 0.

Example

G.f. = 1 - 2*x - x^2 + 2*x^3 + 4*x^5 - 4*x^7 - 4*x^8 - 2*x^9 + 3*x^10 + ...
G.f. = q^2 - 2*q^5 - q^8 + 2*q^11 + 4*q^17 - 4*q^23 - 4*q^26 - 2*q^29 + ...

Mathematica

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

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^12 + A)^2 / (eta(x^6 + A) * eta(x^8 + A)), n))};

Crossrefs

Cf. A127786.

Sequence in context: A340732 A050319 A132456 * A229817 A080966 A187150

Adjacent sequences: A257870 A257871 A257872 * A257874 A257875 A257876

Keyword

sign

Author

Michael Somos, May 11 2015

Status

approved