A186741 Expansion of f(x^5, x^7) in powers of x where f(, ) is Ramanujan's general theta function.

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

Offset

0,1

Comments

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

Links

Seiichi Manyama, 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 24 sequence [ 0, 0, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, ...].

a(n) is the characteristic function of A036498. a(n) = max( 0, A010815(n)).

G.f.: Sum_{k in Z} x^(6*k^2 - k) = Product_{k>0} (1 + x^(12*k - 7)) * (1 + x^(12*k - 5)) * (1 - x^(12*k)).

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

Example

G.f. = 1 + x^5 + x^7 + x^22 + x^26 + x^51 + x^57 + x^92 + x^100 + x^145 + ...
G.f. = q + q^121 + q^169 + q^529 + q^625 + q^1225 + q^1369 + q^2209 + q^2401 + ...

Mathematica

a[n_]:= SeriesCoefficient[ QPochhammer[-q^5, q^12]*QPochhammer[-q^7, q^12] *QPochhammer[q^12, q^12], {q, 0, n}]; (* G. C. Greubel, dec 08 2017 *)

Prog

(PARI) {a(n) = my(m); if( !issquare( 24*n + 1, &m), 0, m%12 == 1 || m%12 == 11)};

Crossrefs

Cf. A010815, A036498, A247223.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A359768 A359787 A247223 * A173864 A353348 A359826

Adjacent sequences: A186738 A186739 A186740 * A186742 A186743 A186744

Keyword

nonn

Author

Michael Somos, Jan 21 2012

Status

approved