A214263 Expansion of f(x^1, x^7) in powers of x where f() is Ramanujan's general theta function.

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

Offset

0,1

Comments

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

Characteristic function of A074377: a(n) = 1 if and only if n is in A074377.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

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

Index entries for characteristic functions.

Formula

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

G.f.: f(x, x^7) = sum_{k in Z} x^(4*k^2 - 3*k).

a(n) = A010054(2*n + 1) = A115359(2*n).

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

Example

G.f. = 1 + x + x^7 + x^10 + x^22 + x^27 + x^45 + x^52 + x^76 + x^85 + x^115 + ...
G.f. = q^9 + q^25 + q^121 + q^169 + q^361 + q^441 + q^729 + q^841 + q^1225 + ...

Mathematica

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; Table[SeriesCoefficient[f[q, q^7], {q, 0, n}], {n, 0, 50}] (* G. C. Greubel, Oct 05 2017 *)

Prog

(PARI) {a(n) = issquare(16*n + 9)};

Crossrefs

Cf. A047621, A074377, A115359.

A000122, A080995, A010054, A133100, A089801 have g.f. of f(x,x^k) for k=1..5.

Cf. A000122, A000700, A121373.

Sequence in context: A359430 A292438 A244525 * A263428 A016355 A016402

Adjacent sequences: A214260 A214261 A214262 * A214264 A214265 A214266

Keyword

nonn,easy

Author

Michael Somos and Omar E. Pol, Jul 09 2012

Status

approved