A274179 Expansion of f(x^1, x^6) in powers of x where f() is Ramanujan's general theta function.

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

Offset

0

Links

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f.: f(x, x^6) = Sum_{k in Z} x^((7k^2 + 5k)/2).

G.f.: Product_{k>0} (1 + x^(7*k-6)) * (1 + x^(7*k-1)) * (1 - x^(7*k)). - Michael Somos, Jan 28 2017

a(3*n + 2) = a(5*n + 2) = a(5*n + 3) = 0. - Michael Somos, Jan 28 2017

a(n) = abs(A232714(n)). - Michael Somos, Jan 28 2017

Example

G.f. = 1 + x + x^6 + x^9 + x^19 + x^24 + x^39 + x^46 + x^66 + x^75 + x^100 + ...
G.f. = q^25 + q^61 + q^241 + q^349 + q^709 + q^889 + q^1429 + q^1681 + q^2401 + ...

Mathematica

a[ n_] := If[ n < 0, 0, Boole @ IntegerQ @ Sqrt @ n]; (* Michael Somos, Jan 28 2017 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^7] QPochhammer[ -x^6, x^7] QPochhammer[ x^7], {x, 0, n}]; (* Michael Somos, Jan 28 2017 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 1, 0, 0, 0, -1, 1, -1, 0, 0, 0, 1, -1, 1}[[Mod[k, 14, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Jan 28 2017 *)

Prog

(PARI) {a(n) = issquare(56*n + 25)}; /* Michael Somos, Jan 28 2017 */
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[1, -1, 1, 0, 0, 0, -1, 1, -1, 0, 0, 0, 1, -1][k%14 + 1], 1 + x * O(x^n)), n))}; /* Michael Somos, Jan 28 2017 */

Crossrefs

Cf. A232714.

Cf. f(x, x^k): A080995 (k=2), A010054 (k=3), A133100 (k=4), A089801(k=5), this sequence (k=6), A214263 (k=7).

Sequence in context: A362130 A361018 A232714 * A194675 A117964 A179020

Adjacent sequences: A274176 A274177 A274178 * A274180 A274181 A274182

Keyword

nonn

Author

Seiichi Manyama, Jun 12 2016

Status

approved