A117410 Expansion of q^(-5/24) * eta(q^2)^3 / eta(q) in powers of q.

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

Offset: 0

Author

Michael Somos, Mar 13 2006

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 + x^13 + ...
G.f. = q^5 + q^29 - q^53 - q^101 - 2*q^125 + q^149 - q^173 - q^197 + q^245 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• B. Gordon and D. Sinor, Multiplicative properties of eta-products, Number theory, Madras 1987, pp. 173-200, Lecture Notes in Math., 1395, Springer, Berlin, 1989. see page 183. MR1019331 (90k:11050).
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A107034.

Programs

Maple

# Uses EulerTransform from A358369.
a := EulerTransform(BinaryRecurrenceSequence(0, 1, -2)):
seq(a(n), n = 0..104); # Peter Luschny, Nov 17 2022

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[x^2]^3 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / eta(x + A), n))};

PARI

q='q+O('q^99); Vec(eta(q^2)^3/eta(q)) \\ Altug Alkan, Apr 17 2018

Sage

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, -2)
a = EulerTransform(b)
print([a(n) for n in range(105)]) # Peter Luschny, Nov 17 2022

Formula

Expansion of psi(x)^2 * chi(-x) = f(-x)^2 / chi(-x)^3 = f(-x)^5 / phi(-x)^3 = f(-x^2)^2 / chi(-x) = f(-x^2)^3 / f(-x) = psi(x) * f(-x^2) = f(x) * f(-x^4) = phi(-x)^2 / chi(-x)^5 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jan 31 2015

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

Given A = A0 + A1 + A2 + A3 + A4 is the 5-section, then 0 = A3 * A1^2 - A2 * A4^2.

Given A = A0 + A1 + A2 + A3 + A4 + A5 + A6 is the 7-section, then 0 = A0*A6 + A1*A5 + A2*A4 + 4*A3^2, A3 = x^10 * A(x^49).

G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2.

A107034(n) = (-1)^n * a(n).