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

Author

N. J. A. Sloane, Feb 02 2012

Keywords

• nonn
• changed

Comments

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

Examples

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

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

Crossrefs

Cf. A036666, A205988.

Programs

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 *)

PARI

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

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