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

0,1

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, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Cf. A010054, A047522, A074378.

Cf. A000122, A000700, A121373.

Sequence in context: A273511 A104015 A244465 * A173858 A217586 A359156

Adjacent sequences: A214261 A214262 A214263 * A214265 A214266 A214267

Keyword

nonn

Author

Michael Somos, Jul 09 2012

Status

approved