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

0,9

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 11th equation.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

Michael Somos, Introduction to Ramanujan theta functions

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: A377069 A261897 A131084 * A219605 A356027 A123949

Adjacent sequences: A143064 A143065 A143066 * A143068 A143069 A143070

Keyword

sign

Author

Michael Somos, Jul 21 2008

Status

approved