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 A116915 Expansion of f(-x, -x^4)^2 / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. 2
 1, -1, 1, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 1, 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, 1, 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, -1, 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 (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). 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^3, b = -x^2. - Michael Somos, Jul 12 2012 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^4, -x^11) - x * f(-x, -x^14) where f() is Ramanujan's two-variable theta function. - Michael Somos, Nov 08 2015 Euler transform of period 5 sequence [ -1, 1, 1, -1, -1, ...]. G.f.: Sum_{k in Z} (-1)^k * (x^((15*k^2 - 7*k)/2) - x^((15*k^2 + 13*k)/2 + 1)). G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 1)) * (1 - x^(5*k - 4)) / ((1 - x^(5*k - 2)) * (1 - x^(5*k - 3))). - a(n) = A010815(5*n + 2). EXAMPLE G.f. = 1 - x + x^2 - x^4 - x^11 + x^15 - x^18 + x^23 + x^37 - x^44 + x^49 - x^57 + ... G.f. = q^49 - q^169 + q^289 - q^529 - q^1369 + q^1849 - q^2209 + q^2809 + q^4489 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ (QPochhammer[ x^5] QPochhammer[ x, x^5] QPochhammer[ x^4, x^5])^2 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jul 12 2012 *) a[ n_] := With[ {k = Sqrt[ 120 n + 49]}, If[ IntegerQ[ k], -KroneckerSymbol[ 12, k], 0]]; (* Michael Somos, Nov 08 2015 *) PROG (PARI) {a(n) = n = 5*n + 2; if( issquare(24*n + 1, &n), -kronecker( 12, n))}; (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))}; CROSSREFS Cf. A010815. Sequence in context: A071003 A071002 A113431 * A076141 A011751 A214507 Adjacent sequences:  A116912 A116913 A116914 * A116916 A116917 A116918 KEYWORD sign AUTHOR Michael Somos, Feb 26 2006 STATUS approved

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Last modified June 21 23:05 EDT 2018. Contains 305646 sequences. (Running on oeis4.)