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 A113681 Expansion of f(-x^2, -x^3)^2 / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function. 3
 1, 1, 0, -1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 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, -1, 0, 0, -1, 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, 1, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). f(a,b) = Sum_{k} a^((k^2+k)/2)*b^((k^2-k)/2) is Ramanujan's two-variable theta function. |a(n)| is the characteristic function of A093722. The exponents in the q-series for this sequence are the squares of the numbers of A057538. 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^4, b = -x. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions Eric Weisstein's World of Mathematics, Quintuple Product Identity FORMULA Expansion of f(-x^7, -x^8) + x * f(-x^2, -x^13) where f() is Ramanujan's two-variable theta function. Euler transform of period 5 sequence [ 1, -1, -1, 1, -1, ...]. G.f.: Sum_{k} (-1)^k * x^(5*k * (3*k + 1)/2) * (x^(-3*k) + x^(3*k + 1)). G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3)) / ((1 - x^(5*k - 1)) * (1 - x^(5*k - 4))). A010815(5*n) = a(n). EXAMPLE 1 + x - x^3 - x^7 - x^8 - x^14 + x^20 + x^29 + x^31 + x^42 - x^52 - x^66 - ... q + q^121 - q^361 - q^841 - q^961 - q^1681 + q^2401 + q^3481 + q^3721 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ (QPochhammer[ q^5] QPochhammer[ q^2, q^5] QPochhammer[ q^3, q^5])^2 / QPochhammer[ q], {q, 0, n}] (* Michael Somos, Jul 17 2012 *) PROG (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^((k%5==0) - kronecker( 5, k)), 1 + x * O(x^n)), n))} (PARI) {a(n) = n*=5; if( issquare( 24*n + 1, &n), kronecker( 12, n))} CROSSREFS Cf. A010815, A057538, A093722, A113430. Sequence in context: A208546 A113430 A214529 * A295895 A179416 A155972 Adjacent sequences:  A113678 A113679 A113680 * A113682 A113683 A113684 KEYWORD sign AUTHOR Michael Somos, Nov 04 2005 STATUS approved

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Last modified July 17 14:44 EDT 2019. Contains 325106 sequences. (Running on oeis4.)