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

Author

Michael Somos and Omar E. Pol, Jul 09 2012

Keywords

• nonn
• easy

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.

Examples

			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 + ...
		

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.

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.

Programs

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 *)

PARI

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

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