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 A143067 Expansion of psi(-x^3) / f(-x^4) in powers of x where psi(), f() are Ramanujan theta functions. 3
 1, 0, 0, -1, 1, 0, 0, -1, 2, -1, 0, -2, 3, -1, 0, -3, 5, -2, 1, -5, 7, -3, 1, -7, 11, -5, 2, -11, 15, -7, 4, -15, 22, -11, 6, -22, 30, -15, 9, -30, 42, -22, 14, -42, 56, -31, 20, -56, 77, -43, 29, -77, 101, -58, 41, -101, 135, -80, 57, -135, 176, -106, 78 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). REFERENCES Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 11th equation. 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 f(x, x^5) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function. Expansion of q^(-5/24) * eta(q^3) * eta(q^12) / (eta(q^4) * eta(q^6)) in powers of q. Euler transform of period 12 sequence [ 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, ...]. G.f.: (1 + x + x^5 + x^8 + x^16 + x^21 + ...) / (1 + x + x^3 + x^6 + x^10 + ...). [Ramanujan] G.f.: 1 - x^3 * (1 - x) / (1 - x^4) + x^8 * (1 - x) * (1 - x^3) / ((1 - x^4) * (1 - x^8)) - ... [Ramanujan] a(2*n) = A262064(n). a(2*n + 3) = - A262090(n). Convolution of A089801 and A106507. - Michael Somos, Jan 10 2017 EXAMPLE G.f. = 1 - x^3 + x^4 - x^7 + 2*x^8 - x^9 - 2*x^11 + 3*x^12 - x^13 - 3*x^15 + ... G.f. = q^5 - q^77 + q^101 - q^173 + 2*q^197 - q^221 - 2*q^269 + 3*q^293 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {x}, {-x^2}, x^2, x^3], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *) a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) EllipticTheta[ 2, Pi/4, x^(3/2)] / QPochhammer[ x^4], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *) a[ n_] := SeriesCoefficient[ x^(-5/24) (EllipticTheta[ 3, 0, x^(1/3)] - EllipticTheta[ 3, 0, x^3]) / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}]; (* Michael Somos, Jan 10 2017 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^12 + A) / (eta(x^4 + A) * eta(x^6 + A)), n))}; CROSSREFS Cf. A089801, A106507, A262064, A262090. Sequence in context: A104402 A261897 A131084 * A219605 A123949 A236358 Adjacent sequences:  A143064 A143065 A143066 * A143068 A143069 A143070 KEYWORD sign AUTHOR Michael Somos, Jul 21 2008 STATUS approved

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Last modified September 21 11:45 EDT 2019. Contains 327253 sequences. (Running on oeis4.)