A185295 a(n) = - A010815(7*n + 1).

1, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 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, 0, 0, 0, 0, 0, 1, 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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0

Offset

0,1

Comments

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

This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^5, b = -x^2.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Eric Weisstein's World of Mathematics, Quintuple Product Identity

Formula

Expansion of f(-x^8, -x^13) + x^2 * f(-x, -x^20) = f(-x^7, -x^14) * f(-x^3, -x^4) / f(-x^2, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.

Euler transform of period 7 sequence [ 0, 1, -1, -1, 1, 0, -1, ...].

Sum_{k} (-1)^k * x^(7*k * (3*k + 1) / 2) * (x^(6*k + 2) + x^(-6*k)).

Product_{k>0} (1 - x^(7*k)) * (1 - x^(7*k - 3)) * (1 - x^(7*k - 4)) / ((1 - x^(7*k - 2)) * (1 - x^(7*k - 5))).

Example

1 + x^2 - x^3 - x^8 - x^13 - x^22 + x^25 + x^37 + x^47 + x^63 - x^68 + ...
q^25 + q^361 - q^529 - q^1369 - q^2209 - q^3721 + q^4225 + q^6241 + q^7921 + ...

Mathematica

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A185295[n_] := SeriesCoefficient[f[-x^7, -x^14]*f[-x^3, -x^4]/ f[-x^2, -x^5], {x, 0, n}]; Table[A185295[n], {n, 0, 50}] (* G. C. Greubel, Jun 19 2017 *)
nmax = 100; CoefficientList[Series[Product[(1 - x^(7*k)) * (1 - x^(7*k-3)) * (1 - x^(7*k-4)) / ((1 - x^(7*k-2)) * (1 - x^(7*k-5))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 20 2017 *)

Prog

(PARI) {a(n) = local(m); if( issquare( 168*n + 25, &m), - kronecker( 12, m))}

Crossrefs

Cf. A010815.

Sequence in context: A258769 A266377 A266326 * A214295 A145377 A246260

Adjacent sequences: A185292 A185293 A185294 * A185296 A185297 A185298

Keyword

sign

Author

Michael Somos, Jan 21 2012

Status

approved