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A185295
a(n) = - A010815(7*n + 1).
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
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 A374113
KEYWORD
sign
AUTHOR
Michael Somos, Jan 21 2012
STATUS
approved