A258741 Expansion of f(x^3, x^5) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

1, -1, 1, -1, 2, -2, 2, -3, 4, -5, 5, -6, 8, -9, 10, -12, 15, -17, 19, -22, 26, -30, 33, -38, 45, -51, 56, -64, 74, -83, 92, -104, 119, -133, 147, -165, 187, -208, 229, -256, 288, -319, 351, -390, 435, -481, 528, -584, 649, -715, 783, -863, 954, -1047, 1145

Offset: 0

Author

Michael Somos, Nov 06 2015

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - x + x^2 - x^3 + 2*x^4 - 2*x^5 + 2*x^6 - 3*x^7 + 4*x^8 - 5*x^9 + ...
G.f. = 1/q - q^15 + q^31 - q^47 + 2*q^63 - 2*q^79 + 2*q^95 - 3*q^111 + ...
		

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 16th equation.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. A029838, A036016, A082303, A106507, A214264.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] / (QPochhammer[ -x, x^8] QPochhammer[ -x^7, x^8]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ 1 / Product[ (1 + x^(8 k + 1)) (1 - x^(8 k + 4)) (1 + x^(8 k + 7)), {k, 0, Ceiling[ n/8]}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 1, 0}[[Mod[k, 16, 1]]], {k, n}], {x, 0, n}];

PARI

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 0, 1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 1][k%16 + 1]), n))};

Formula

Expansion of f(x^4, x^12) / f(x, x^7) where f(, ) is Ramanujan's general theta function.

Euler transform of period 16 sequence [ -1, 1, 0, 1, 0, 0, -1, 0, -1, 0, 0, 1, 0, 1, -1, 0, ...].

G.f.: 1 / (Product_{k>=0} (1 + x^(8*k + 1)) * (1 - x^(8*k + 4)) * (1 + x^(8*k + 7))).

G.f.: (1 + x^4 + x^12 + x^24 + x^40 + ...) / (1 + x + x^7 + x^10 + x^22 + ...). [Ramanujan]

G.f.: 1 - x * (1 - x) / (1 - x^2) + x^4 * (1 - x) * (1 - x^3) / ((1 - x^2) * (1 - x^4)) - ... [Ramanujan]

a(n) = (-1)^n * A036016(n) = A029838(2*n) = A082303(2*n).

Convolution product of A106507 and A214264.