The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A207710 Expansion of f(x) * f(-x^10) / f(-x^2, -x^8) in powers of x where f() is Ramanujan's two-variable theta function. 1
 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 (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^4, b = -x. 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^5, -x^10) * f(-x^2, x^3) / f(-x, x^4) = f(-x^7, x^8) + x * f(x^2, -x^13) in powers of x where f() is Ramanujan's two-variable theta function. Euler transform of period 20 sequence [ 1, -1, 1, -1, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, -1, 1, -1, 1, -1, ...]. |a(n)| is the characteristic function of A093722. The exponents in the q-series q * A(q^120) are the squares of the numbers in A057538. G.f.: Prod_{k>0} (1 - (-x)^k) / ((1 - x^(10*k - 2)) * (1 - x^(10*k - 8))). G.f.: Sum_{k} (-1)^[-k/2] * x^(5*k * (3*k + 1)/2) * (x^(-3*k) + x^(3*k + 1)). a(7*n + 2) = a(7*n + 4) = a(7*n + 5) = 0. a(n) * (-1)^n = A113430(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 f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A207710[n_] := SeriesCoefficient[f[x^5, -x^10]*f[-x^2, x^3]/f[-x, x^4], {x, 0, n}]; Table[A207710[n], {n, 0, 50}] (* G. C. Greubel, Jun 18 2017 *) PROG (PARI) {a(n) = local(m); if( issquare( 120*n + 1, &m), kronecker( -120, m) * (-1)^(m \ 15))} CROSSREFS Cf. A057538, A093722, A113430. Sequence in context: A214509 A053866 A143259 * A207735 A208546 A113430 Adjacent sequences:  A207707 A207708 A207709 * A207711 A207712 A207713 KEYWORD sign AUTHOR Michael Somos, Feb 19 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 1 09:26 EST 2021. Contains 349426 sequences. (Running on oeis4.)