A245433 Expansion of f(-x^3, -x^5)^2 / (psi(-x) * psi(x^2)) in powers of x where psi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta functions.

1, 1, 0, -1, 1, 1, -2, -2, 3, 4, -4, -5, 5, 6, -8, -9, 12, 13, -14, -17, 18, 21, -26, -28, 34, 39, -42, -49, 53, 60, -70, -78, 90, 101, -110, -125, 137, 153, -174, -192, 217, 241, -264, -295, 322, 357, -400, -438, 490, 540, -588, -652, 711, 781, -866, -946

Offset: 0

Author

Michael Somos, Jul 21 2014

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^3 + x^4 + x^5 - 2*x^6 - 2*x^7 + 3*x^8 + 4*x^9 + ...
G.f. = 1/q + q^3 - q^11 + q^15 + q^19 - 2*q^23 - 2*q^27 + 3*q^31 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A111374, A245436.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 1, 1, -2, 1, 1, -1, 0}[[Mod[k, 8, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Jun 27 2017 *)
f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a:= CoefficientList[Series[f[-x^3, -x^5]^2/(f[-x, -x^3]*f[x^2, x^6]), {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 06 2018 *)

PARI

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -1, 1, 1, -2, 1, 1, -1][k%8 + 1]), n))};

Formula

Euler transform of period 8 sequence [1, -1, -1, 2, -1, -1, 1, 0, ...].

Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - v)^3 - 4 * u^2 * v^3 * (2*v - u^2) * (u^2*v - v^2 - 2).

a(n) = A111374(2*n) = A245436(4*n - 1).