A246927 Expansion of psi(-q) * phi(q^3)^2 * chi(q^3) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

1, -1, 0, 4, -5, 0, 4, -8, 0, 2, -4, 0, 12, -4, 0, 16, -13, 0, 0, -12, 0, 8, -4, 0, 20, -5, 0, 4, -16, 0, 8, -24, 0, 8, -8, 0, 10, -4, 0, 32, -20, 0, 8, -12, 0, 0, -8, 0, 28, -9, 0, 24, -20, 0, 4, -32, 0, 8, -4, 0, 32, -12, 0, 16, -29, 0, 16, -12, 0, 16, -8

Offset

0,4

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

Euler transform of period 12 sequence [-1, 0, 4, -1, -1, -7, -1, -1, 4, 0, -1, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 246^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246926.

a(3*n) = A034933(n). a(3*n + 1) = - A246926(n). a(3*n + 2) = 0.

Example

G.f. = 1 - q + 4*q^3 - 5*q^4 + 4*q^6 - 8*q^7 + 2*q^9 - 4*q^10 + 12*q^12 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3, q^6] EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 3, 0, q^3]^2 / (2^(1/2) q^(1/8)), {q, 0, n}];

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^12 / (eta(x^2 + A) * eta(x^3 + A)^5 * eta(x^12 + A)^5), n))};
(Magma) A := Basis( ModularForms( Gamma0(36), 3/2), 70); A[1] - A[2] + 4*A[4] - 5*A[5] + 4*A[6];

Crossrefs

Cf. A034933, A246926.

Sequence in context: A197584 A213440 A011286 * A187045 A186930 A159567

Adjacent sequences: A246924 A246925 A246926 * A246928 A246929 A246930

Keyword

sign

Author

Michael Somos, Sep 07 2014

Status

approved