A186742 Expansion of f(x, x^11) in powers of x where f(, ) is Ramanujan's general theta function.

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

Offset: 0

Author

Michael Somos, Jan 21 2012

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + x + x^11 + x^14 + x^34 + x^39 + x^69 + x^76 + x^116 + x^125 + ...
G.f. = q^25 + q^49 + q^289 + q^361 + q^841 + q^961 + q^1681 + q^1849 + ...
		

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.

Crossrefs

Cf. A010815, A195818.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[ n_] := With[{m = Sqrt[24 n + 25]}, If[ n >= 0 && IntegerQ @ m, Boole[ Mod[m, 12] == 5 || Mod[m, 12] == 7], 0]]; (* Michael Somos, Aug 28 2017 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^12] QPochhammer[ -x^1, x^12] QPochhammer[ -x^11, x^12], {x, 0, n}]; (* Michael Somos, Aug 28 2017 *)

PARI

{a(n) = my(m); n++; if( !issquare( 24*n + 1, &m), 0, m%12 == 5 || m%12 == 7)};

PARI

{a(n) = my(A); n = 3*n + 3; if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A) - eta(x^3 + A) - x * eta(x^18 + A)^2 / eta(x^9 + A)) / 2, n))}; /* Michael Somos, Aug 28 2017 */

Formula

Euler transform of period 24 sequence [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, ...].

a(n) is the characteristic function of A195818. a(n) = max( 0, -A010815(n + 1)).

G.f.: Sum_{k in Z} x^(6*k^2 - 5*k) = Product_{k>0} (1 + x^(12*k - 11)) * (1 + x^(12*k - 1)) * (1 - x^(12*k)).

Expansion of (f(x, x^2) - f(-x, -x^2)) / (2*x) in powers of x. - Michael Somos, Aug 28 2017

Sum_{k=1..n} a(k) ~ sqrt(2*n/3). - Amiram Eldar, Jan 13 2024