A254372 Expansion of phi(q) * phi(-q^3) * f(-q^12) / f(-q^4)^3 in powers of q where phi(), f() are Ramanujan theta functions.

1, 2, 0, -2, 1, 6, 0, -10, 3, 20, 0, -30, 1, 52, 0, -78, 6, 126, 0, -184, 3, 280, 0, -402, 12, 590, 0, -830, 5, 1182, 0, -1636, 21, 2280, 0, -3108, 10, 4252, 0, -5722, 36, 7710, 0, -10252, 15, 13632, 0, -17940, 60, 23586, 0, -30744, 26, 40014, 0, -51714, 96

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 eta(q^2)^5 * eta(q^3)^2 * eta(q^12) / (eta(q)^2 * eta(q^4)^5 * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [ 2, -3, 0, 2, 2, -4, 2, 2, 0, -3, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = (4/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A230256.

a(4*n + 2) = 0. a(2*n + 1) = 2 * A254346(n). a(4*n) = A132180(n).

Example

G.f. = 1 + 2*q - 2*q^3 + q^4 + 6*q^5 - 10*q^7 + 3*q^8 + 20*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] * EllipticTheta[ 4, 0, q^3] QPochhammer[ q^12] / QPochhammer[ q^4]^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)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A)^5 * eta(x^6 + A)), n))};

Crossrefs

Cf. A132180, A254346, A230256.

Sequence in context: A049785 A036997 A116900 * A354158 A363580 A196517

Adjacent sequences: A254369 A254370 A254371 * A254373 A254374 A254375

Keyword

sign

Author

Michael Somos, Jan 29 2015

Status

approved