A143067 Expansion of psi(-x^3) / f(-x^4) in powers of x where psi(), f() are Ramanujan theta functions.

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

Offset: 0

Author

Michael Somos, Jul 21 2008

Keywords

• sign

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			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 + ...
		

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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 11th equation.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
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A089801, A106507, A262064, A262090.

Programs

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 *)

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))};

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