A185124 Expansion of f(x, -x^5) in powers of x where f(,) is the Ramanujan general theta function.

1, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,1

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

a(n) is nonzero if and only if n is a number of A001082.

The exponents in the q-series for this sequence are the squares of the numbers of A001651.

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

Euler transform of period 24 sequence [ 1, -1, 0, 0, -1, 1, -1, 0, 0, 0, 1, -2, 1, 0, 0, 0, -1, 1, -1, 0, 0, -1, 1, -1, ...].

G.f.: Sum_{k in Z} (-1)^floor(k + 1)/2) * x^(k * (3*k + 2)).

a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 4) = a(8*n + 4) = 0. a(4*n + 1) = A080902(n). a(8*n) = A010815(n).

a(n) = (-1)^n * A185125(n). - Michael Somos, Jun 30 2015

Example

G.f. = 1 + x - x^5 - x^8 - x^16 - x^21 + x^33 + x^40 + x^56 + x^65 - x^85 + ...
G.f. = q + q^4 - q^16 - q^25 - q^49 - q^64 + q^100 + q^121 + q^169 + q^196 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, -x^6] QPochhammer[ x^5, -x^6] QPochhammer[ -x^6], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)

Prog

(PARI) {a(n) = my(m); if( issquare( 3*n + 1, &m), (m%3!=0) * (-1)^((m+3) \ 6), 0)};

Crossrefs

Cf. A010815, A080902, A185125.

Sequence in context: A089801 A290739 A143064 * A185125 A327580 A163811

Adjacent sequences: A185121 A185122 A185123 * A185125 A185126 A185127

Keyword

sign

Author

Michael Somos, Jan 20 2012

Status

approved