A226559 Expansion of f(-x^1, -x^7) * f(-x^2, -x^6) / (f(-x^3, -x^5) * f(-x^4, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function.

1, -1, -1, 2, 1, -2, -2, 2, 4, -4, -5, 6, 6, -7, -9, 10, 13, -15, -17, 20, 21, -25, -28, 32, 39, -43, -49, 56, 60, -69, -78, 86, 101, -112, -125, 142, 153, -172, -192, 212, 241, -266, -295, 328, 357, -397, -438, 482, 540, -592, -652, 720, 781, -862, -946

Offset: 0

Author

Michael Somos, Jun 10 2013

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 + 2*x^3 + x^4 - 2*x^5 - 2*x^6 + 2*x^7 + 4*x^8 - 4*x^9 + ...
G.f. = q^3 - q^7 - q^11 + 2*q^15 + q^19 - 2*q^23 - 2*q^27 + 2*q^31 + 4*q^35 + ...
		

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. A092869, A230534.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{0, 1, 1, -1, -2, -1, 1, 1}[[ Mod[k, 8] + 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, -1, -2, -1, 1, 1][k%8 + 1]), n))};

Formula

Expansion of (f(-x^1, -x^7) / f(-x^3, -x^5)) * (psi(-x^2) / phi(-x^4)) in powers of x where psi(), phi(), f() are Ramanujan theta functions.

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

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

a(n) = -A092869(2*n + 1) = A230534(2*n + 1).