A134014 Expansion of phi(-q) * phi(q^4) in powers of q where phi() is a Ramanujan theta function.

1, -2, 0, 0, 4, -4, 0, 0, 4, -2, 0, 0, 0, -4, 0, 0, 4, -4, 0, 0, 8, 0, 0, 0, 0, -6, 0, 0, 0, -4, 0, 0, 4, 0, 0, 0, 4, -4, 0, 0, 8, -4, 0, 0, 0, -4, 0, 0, 0, -2, 0, 0, 8, -4, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 4, -8, 0, 0, 8, 0, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 8, -2

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 * eta(q^8)^5 / (eta(q^2) * eta(q^4)^2 * eta(q^16)^2) in powers of q.

Euler transform of period 16 sequence [ -2, -1, -2, 1, -2, -1, -2, -4, -2, -1, -2, 1, -2, -1, -2, -2, ...].

Moebius transform is period 16 sequence [ -2, 2, 2, 4, -2, -2, 2, 0, -2, 2, 2, -4, -2, -2, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134013.

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

G.f.: 1 - 2 * ( x / (1 + x^2) + x^3 / (1 + x^6) - 2 * x^4 / (1 + x^8) + ... ).

a(n) = (-1)^n * A004531(n). a(n) = -2 * A134015(n) unless n=0. a(4*n) = A004018(n). a(4*n+1) = - A004020(n).

Example

G.f. = 1 - 2*q + 4*q^4 - 4*q^5 + 4*q^8 - 2*q^9 - 4*q^13 + 4*q^16 - 4*q^17 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^4], {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, if( n%4 < 2, (n%2*-6 + 4) * sumdiv(n, d, kronecker(-4, d))))};
(PARI) {a(n) = (-1)^n * if( n<1, n==0, 2 * qfrep([1, 0; 0, 4], n)[n])};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A)^5 / eta(x^2 + A) / eta(x^4 + A)^2 / eta(x^16 + A)^2, n))};

Crossrefs

Cf. A004018, A004020, A004531, A134015.

Sequence in context: A185146 A080964 A367054 * A004531 A072071 A329264

Adjacent sequences: A134011 A134012 A134013 * A134015 A134016 A134017

Keyword

sign

Author

Michael Somos, Oct 02 2007

Status

approved