A214264 Expansion of f(x^3, x^5) in powers of x where f() is Ramanujan's two-variable theta function.

1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Michael Somos, Jul 09 2012

Keywords

• nonn

Comments

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

Examples

			1 + x^3 + x^5 + x^14 + x^18 + x^33 + x^39 + x^60 + x^68 + x^95 + x^105 +
q + q^49 + q^81 + q^225 + q^289 + q^529 + q^625 + q^961 + q^1089 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A010054, A047522, A074378.

Cf. A000122, A000700, A121373.

Programs

Mathematica

f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A214264[n_] := SeriesCoefficient[f[x^3, x^5], {x, 0, n}]; Table[A214264[n], {n, 0, 50}] (* G. C. Greubel, Dec 03 2017 *)

PARI

{a(n) = issquare( 16*n + 1)}

Formula

Euler transform of period 16 sequence [ 0, 0, 1, 0, 1, -1, 0, -1, 0, -1, 1, 0, 1, 0, 0, -1, ...].

G.f.: Sum_{k} x^(((8*k + 1)^2 - 1) / 16).

Characteristic function of A074378. a(n) = 1 if and only if n is in A074378.

a(n) = A010054(2*n).

Sum_{k=1..n} a(k) ~ sqrt(n). - Amiram Eldar, Jan 13 2024