A263452 Expansion of f(-q^3)^3 * psi(q^12) / f(-q) in powers of q where ps(), f() are Ramanujan theta functions.

1, 1, 2, 0, 2, 1, 2, 0, 1, 2, 2, 0, 3, 1, 4, 0, 5, 3, 2, 0, 3, 3, 4, 0, 4, 2, 4, 0, 3, 2, 4, 0, 4, 2, 4, 0, 5, 5, 4, 0, 3, 3, 8, 0, 7, 3, 6, 0, 4, 4, 4, 0, 6, 4, 4, 0, 9, 3, 6, 0, 4, 4, 4, 0, 4, 3, 8, 0, 5, 5, 6, 0, 9, 3, 4, 0, 7, 6, 6, 0, 7, 6, 10, 0, 6, 3

Offset

0,3

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^(-11/6) * eta(q^3)^3 * eta(q^24)^2 / (eta(q) * eta(q^12)) in powers of q.

Euler transform of period 24 sequence [ 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -1, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -3, ...].

2 * a(n) = A261444(2*n + 1). a(4*n + 1) = A212907(n). a(4*n + 3) = 0.

-2 * a(n) = A263527(2*n + 3). - Michael Somos, Nov 05 2015

Example

G.f. = 1 + x + 2*x^2 + 2*x^4 + x^5 + 2*x^6 + x^8 + 2*x^9 + 2*x^10 + ...
G.f. = q^11 + q^17 + 2*q^23 + 2*q^35 + q^41 + 2*q^47 + q^59 + 2*q^65 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q^3]^3 EllipticTheta[ 2, 0, q^6] / ( 2 q^(3/2) QPochhammer[ q]), {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^24 + A)^2 / (eta(x + A) * eta(x^12 + A)), n))};

Crossrefs

Cf. A212907, A261444, A263527.

Sequence in context: A219762 A227696 A033687 * A133457 A324120 A218857

Adjacent sequences: A263449 A263450 A263451 * A263453 A263454 A263455

Keyword

nonn

Author

Michael Somos, Oct 18 2015

Status

approved