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A185059
a(n) = A010815(7*n).
1
1, 1, 0, 0, 0, -1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 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, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 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^6, b = -x.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Quintuple Product Identity
FORMULA
Expansion of f(-x^10, -x^11) + x * f(-x^4, -x^17) = f(-x^7, -x^14) * f(-x^2, -x^5) / f(-x, -x^6) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 7 sequence [ 1, -1, 0, 0, -1, 1, -1, ...].
Sum_{k} (-1)^k * x^(7*k * (3*k + 1) / 2) * (x^(3*k + 1) + x^(-3*k)).
Product_{k>0} (1 - x^(7*k)) * (1 - x^(7*k - 2)) * (1 - x^(7*k - 5)) / ((1 - x^(7*k - 1)) * (1 - x^(7*k - 6))).
EXAMPLE
1 + x - x^5 - x^10 - x^11 - x^18 + x^30 + x^41 + x^43 + x^56 - x^76 + ...
q + q^169 - q^841 - q^1681 - q^1849 - q^3025 + q^5041 + q^6889 + q^7225 + ...
MATHEMATICA
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A185059[n_] := SeriesCoefficient[f[-x^7, -x^14]*f[-x^2, -x^5]/ f[-x, -x^6], {x, 0, n}]; Table[A185059[n], {n, 0, 50}] (* G. C. Greubel, Jun 19 2017 *)
nmax = 100; CoefficientList[Series[Product[(1 - x^(7*k)) * (1 - x^(7*k-2)) * (1 - x^(7*k-5)) / ((1 - x^(7*k-1)) * (1 - x^(7*k-6))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 20 2017 *)
PROG
(PARI) {a(n) = local(m); if( issquare( 168*n + 1, &m), kronecker( 12, m))}
CROSSREFS
Cf. A010815.
Sequence in context: A256436 A253903 A255849 * A179776 A360130 A353570
KEYWORD
sign
AUTHOR
Michael Somos, Jan 21 2012
STATUS
approved