A107034 Expansion of f(-x) * f(-x^4) in powers of x where f() is a Ramanujan theta function.

1, -1, -1, 0, -1, 2, 1, 1, -1, 0, 1, -1, -1, -1, 0, -2, 1, 0, 0, 1, 2, -1, 0, 1, 0, 1, 0, 1, 1, -1, -3, 0, -1, 1, -1, -1, 0, 0, 0, 1, -2, 0, 1, 0, 1, 0, 1, 0, 0, 1, 2, 1, 0, -1, 1, -3, 0, 1, 0, -1, -1, 0, 1, 0, 0, -2, 0, -1, -1, 0, -2, 1, 1, 0, 0, 1, 0, 0, 1

Offset: 0

Author

Michael Somos, May 09 2005

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 - x^4 + 2*x^5 + x^6 + x^7 - x^8 + x^10 - x^11 - x^12 - ...
G.f. = q^5 - q^29 - q^53 - q^101 + 2*q^125 + q^149 + q^173 - q^197 + q^245 + ...
		

References

• H. Kahl, G. Koehler, Components of Hecke theta series, J. Math. Anal. Appl. 232 (1999), no. 2, 312-331, see page 320. MR1683136 (2000e:11051)

Links

• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^4], {x, 0, n}]; (* Michael Somos, Jan 29 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A), n))};

Formula

Expansion of f(x)^2 / chi(x)^3 = f(x)^5 / phi(x)^3 = f(-x^2)^2 / chi(x) = f(-x^2) * psi(-x) = f(-x^2)^3 / f(x) = phi(x)^2 / chi(x)^5 = psi(-x)^2 * chi(x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jan 29 2015

Expansion of q^(-5/24) * eta(q) * eta(q^4) in powers of q.

Euler transform of period 4 sequence [-1, -1, -1, -2, ...].

G.f. Product_{k>0} (1 - x^k) * (1 - x^(4*k)).