A244525 Expansion of f(-x^1, -x^7) in powers of x where f(, ) is Ramanujan's general theta function.

1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Michael Somos, Jun 29 2014

Keywords

• sign

Examples

			G.f. = 1 - x - x^7 + x^10 + x^22 - x^27 - x^45 + x^52 + x^76 - x^85 + ...
G.f. = q^9 - q^25 - q^121 + q^169 + q^361 - q^441 - q^729 + q^841 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• C. G. J. Jacobi, Uber die Zur Numerischen Berechnung Der Elliptischen Funtionen Zweckmassigsten Formeln, in Gesammelte Werke, Bd. I, 1881, pp. 343-368. See p. 347 equ. (7.)
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A214263.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^1, x^8] QPochhammer[ x^7, x^8] QPochhammer[ x^8], {x, 0, n}];

PARI

{a(n) = issquare(16*n + 9) * (-1)^n};

Formula

Euler transform of period 8 sequence [-1, 0, 0, 0, 0, 0, -1, -1, ...].

G.f.: f(-x, -x^7) = Sum_{k in Z} (-1)^k * x^(4*k^2 - 3*k).

a(n) = (-1)^n * A214263(n).

G.f.: Product_{k>0} (1 - x^(8*k-1)) * (1 - x^(8*k-7)) * (1 - x^(8*k)). - Seiichi Manyama, Jun 14 2016