A259774 Expansion of f(x, x^7) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

1, 0, 0, -1, 1, -1, 1, -1, 2, -2, 2, -3, 4, -4, 4, -6, 7, -7, 8, -10, 12, -13, 14, -17, 21, -22, 24, -29, 33, -36, 40, -46, 53, -58, 63, -73, 83, -90, 99, -113, 127, -138, 152, -171, 191, -209, 228, -255, 285, -309, 338, -377, 416, -453, 495, -547, 603, -656

Offset: 0

Author

Michael Somos, Nov 08 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^3 + x^4 - x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 + 2*x^10 - 3*x^11 + ...
G.f. = q^7 - q^55 + q^71 - q^87 + q^103 - q^119 + 2*q^135 - 2*q^151 + ...
		

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 15th equation.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, A036015, A082303, A106507, A214263.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] / (QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ 1 / Product[ (1 + x^(8 k + 3)) (1 - x^(8 k + 4)) (1 + x^(8 k + 5)), {k, 0, Ceiling[ n/8]}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0, 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, 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0][k%16 + 1]), n))};

Formula

Expansion of f(x^4, x^12) / f(x^3, x^5) where f(, ) is Ramanujan's general theta function.

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

G.f.: (1 + x^4 + x^12 + x^24 + x^40 + ...) / (1 + x^3 + x^5 + x^14 + x^18 + ...). [Ramanujan]

G.f.: 1 - x^3 * (1 - x) / (1 - x^2) + x^8 * (1 - x) * (1 - x^3) / ((1 - x^2) * (1 - x^4)) - ... [Ramanujan]

a(n) = (-1)^n * A036015(n) = A029838(2*n + 1) = - A082303(2*n + 1).

Convolution product of A106507 and A214263.