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 A145708 Expansion of psi(-q) / psi(-q^5) in powers of q where psi() is a Ramanujan theta function. 3
 1, -1, 0, -1, 0, 1, 0, 0, -1, 0, 2, 0, 0, -1, 0, 2, -1, 0, -2, 0, 3, -2, 0, -3, 0, 5, -2, 0, -3, 0, 6, -2, 0, -4, 0, 8, -3, 0, -6, 0, 11, -5, 0, -8, 0, 14, -6, 0, -10, 0, 18, -6, 0, -12, 0, 22, -9, 0, -16, 0, 28, -13, 0, -21, 0, 36, -14, 0, -25, 0, 44, -16, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,11 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of q^(1/2) * eta(q) * eta(q^4) * eta(q^10) / (eta(q^2) * eta(q^5) * eta(q^20)) in powers of q. Euler transform of period 20 sequence [ -1, 0, -1, -1, 0, 0, -1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, 0, -1, 0, ...]. G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A036026. a(5*n + 2) = a(5*n + 4) = 0. G.f.: (Product_{k>0} P(5, x^k) * P(20, x^k))^(-1) where P(n, x) is the n-th cyclotomic polynomial. a(n) = (-1)^n * A138532(n). a(5*n + 3) = - A036026(n). Convolution square is A145740. Convolution inverse is A036026. a(n) = A145723(2*n - 1). a(2*n) = A146164(n). a(2*n + 1) = - A147699(n). - Michael Somos, Sep 06 2015 EXAMPLE G.f. = 1 - x - x^3 + x^5 - x^8 + 2*x^10 - x^13 + 2*x^15 - x^16 - 2*x^18 + ... G.f. = 1/q - q - q^5 + q^9 - q^15 + 2*q^19 - q^25 + 2*q^29 - q^31 - 2*q^35 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ x^(1/2) EllipticTheta[ 2, Pi/4, x^(1/2)] / EllipticTheta[ 2, Pi/4, x^(5/2)], {x, 0, n}]; (* Michael Somos, Sep 06 2015 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A) / (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))}; CROSSREFS Cf. A036026, A138532, A145723, A145740, A146164. Sequence in context: A323882 A084143 A025888 * A138532 A065293 A164615 Adjacent sequences:  A145705 A145706 A145707 * A145709 A145710 A145711 KEYWORD sign AUTHOR Michael Somos, Oct 17 2008 STATUS approved

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Last modified January 20 11:11 EST 2020. Contains 331083 sequences. (Running on oeis4.)