A108483 Expansion of f(-x^2, -x^5) / f(-x, -x^6) in powers of x where f() is a Ramanujan theta function.

1, 1, 0, 0, 0, -1, 0, 1, 1, 0, -1, -1, -1, 0, 2, 2, 0, -1, -2, -2, 0, 3, 3, 0, -2, -3, -3, 0, 5, 5, 1, -3, -5, -5, 0, 7, 7, 1, -5, -8, -7, 1, 11, 12, 2, -7, -12, -11, 1, 15, 16, 3, -11, -18, -15, 2, 23, 24, 5, -15, -26, -22, 3, 31, 33, 7, -22, -37, -30, 5, 44, 47, 11, -30, -52, -42, 6, 59, 63, 15, -42, -72, -56, 10, 82, 88, 22

Offset: 0

Author

Michael Somos, Jun 04 2005

Keywords

• sign

Comments

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

In Duke (2005) page 157 the g.f. is denoted by t(tau).

Examples

			G.f. = 1 + x - x^5 + x^7 + x^8 - x^10 - x^11 - x^12 + 2*x^14 + 2*x^15 + ...
G.f. = q^-2 + q^5 - q^33 + q^47 + q^54 - q^68 - q^75 - q^82 + 2*q^96 + ...
		

References

• W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162.

Links

• Michael Somos, Introduction to Ramanujan theta functions [This is just section 1 of the next item, but it is mentioned in over 1000 sequences. - _N. J. A. Sloane_, Nov 13 2019]
• Michael Somos, A Multisection of q-Series
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A229894, A375106, A375150.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^7] QPochhammer[ x^5, x^7] / (QPochhammer[ x, x^7] QPochhammer[ x^6, x^7]), {x, 0, n}]; (* Michael Somos, Oct 03 2013 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 1, 0, 0, 1, -1, 0}[[Mod[k, 7, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 03 2015 *)

PARI

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x*O(x^n))^[ 0, -1, 1, 0, 0, 1, -1][k%7 + 1]), n))};

Formula

Euler transform of period 7 sequence [ 1, -1, 0, 0, -1, 1, 0, ...]. - Michael Somos, Oct 03 2013

Given g.f. A(x), then B(q) = q^-2*A(q^7) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^3 - u^6 + 3*u^4*v + u^7*v^3 + u^2*v^9 + u^8*v^6 - 3*u^2*v^2 - 2*u*v^6 - 5*u^3*v^5 - u^5*v^4 - u^9*v^2 - u^4*v^8 - u^6*v^7.

G.f.: Product_{k>0} (1 - x^(7*k - 2)) * (1 - x^(7*k - 5)) / ((1 - x^(7*k - 1)) * (1 - x^(7*k - 6))).

a(n) = A229894(7*n). - Michael Somos, Oct 03 2013

G.f.: B(x) / C(x), where B(x) is the g.f. of A375106 and C(x) is the g.f. of A375150. - Seiichi Manyama, Aug 03 2024