A205633 Expansion of f(x^3, x^7) in powers of x where f() is Ramanujan's two-variable theta function.

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0, 1, 0, 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, 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, 0, 0, 0, 0, 0, 0, 0, 1, 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).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See D(q).

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 20 sequence [ 0, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0, 0, -1, ...]. - Michael Somos, Sep 22 2012

Characteristic function of A036666: a(n) = 1 if and only if n is in A036666.

G.f.: Sum_{k in Z} x^(5*k^2 + 2*k).

a(3*n + 2) = a(4*n + 1) = a(4*n + 2) = 0. a(4*n + 3) = A205988(n). - Michael Somos, Sep 22 2012

Example

1 + x^3 + x^7 + x^16 + x^24 + x^39 + x^51 + x^72 + x^88 + x^115 + x^135 + ...
q + q^16 + q^36 + q^81 + q^121 + q^196 + q^256 + q^361 + q^441 + q^576 + ...

Mathematica

A205633[n_] := SeriesCoefficient[QPochhammer[-q^3, q^10]* QPochhammer[-q^7, q^10]*QPochhammer[q^10, q^10], {q, 0, n}]; Table[A205633[n], {n, 0, 50}] (* G. C. Greubel, Jun 16 2017 *)

Prog

(PARI) {a(n) = issquare(5*n + 1)} /* Michael Somos, Sep 22 2012 */

Crossrefs

Cf. A036666, A205988.

Sequence in context: A359835 A353331 A353269 * A352594 A252488 A170956

Adjacent sequences: A205630 A205631 A205632 * A205634 A205635 A205636

Keyword

nonn

Author

N. J. A. Sloane, Feb 02 2012

Status

approved