OEIS/Eta products

Eta product signatures (eps)

The coefficient sequences for generating functions that are products of the eta function can be computed conveniently by an Euler transform of some periodic integer sequence. Following Michael Somos, we describe an eta product by a matrix resp. a list of pairs (qi, ei) of the form

[q1,e1;q2,e2;q3,e3;...]

The q_i the powers of q and the e_i are powers of the eta function. The latter can be negative.

Partition numbers

A000041 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231
Euler transform of period 1: [1]

Dedkind eta η (without the q^(1/24) factor)

A010815 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1
eta(q)
Euler transform of period 1: [-1]
eps P="[1,1]"

Jacobi theta_2 θ_2

A089800 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2 ... = theta_2(q)/q^(1/4)

Jacobi theta_3 θ_3, Ramanujan phi φ

A000122 1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2
a(0) = 1, for n >= 1: a(n) = 2 if n is a square, otherwise 0.
eta(q^2)^5 / (eta(q)*eta(q^4))^2
Euler transform of period 4: [2,-3,2,-1]
eps P="[2,5;1,-2;4,-2]"

Jacobi theta_4 θ_4

A002448 1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2
a(0) = 1, for n >= 1: a(n) = 2 * (-1)^sqrt(n) if n is a square, otherwise 0.
eta(q)^2 / eta(q^2)
Euler transform of period 2: [2,-1]
eps P="[1,2;2,-1]"

Ramanujan psi ψ

A010054 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0
a(n) = 1 if n is a triangular number, otherwise 0.
q^(-1/8) * eta(q^2)^2 / eta(q)
Euler transform of period 2: [1,-1]
eps P="[2,2;1,-1]"

Ramanujan chi χ

A000700 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5
q^(1/24) * eta(q^2)^2 /(eta(q) * eta(q^4))
Euler transform of period 4: [1,-1,1,0]
eps P="[2,2;1,-1;4,-1]"

Ramanujan f

A121373 1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1
a(n) = (-1)^n * A010815(n)
q^(-1/24) * eta(q^2)^3 / (eta(q) * eta(q^4))
Euler transform of period 4: [[1,-2,1,-1]
eps P="[2,3;1,-1;4,-1]"

Leading power of q factor (pqf)

Some of the generating functions have an additional power of q factor, in order to normalize the leading coefficient to q^0.

• Example A226862:

Expansion of q^(-1/6) * eta(q^4) * eta(q^6)^5 / (eta(q^3) * eta(q^12))^2
eps P="[4,1;6,5;3,-2;12,-2]"

For all pairs we multiply the q power with the eta power and add the results, giving 4*1 + 6*5 - 3*2 - 12*2 = 4 + 30 - 6 - 24 = 4. This is the negative numerator of the pqf (expanded for denominator 24).

• Example A286813:

q^(-9/8) * (eta(q^2) * eta(q^16))^2 / (eta(q) * eta(q^8))
eps P="[2,2;16,2;1,-1;8,-1]" gives 2*2 * 16*2  - 1*1 - 8*1 = 36 - 9 = 27 and pqf = q^(-27/24)

• Example A286134 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, -1, -1

q^(-1/2) * eta(q^5) * eta(q^6) * eta(q^7) * eta(q^210)
eps P="[5,1;6,1;7,1;210,1]" gives 18 + 210 = 228 and pqf = q^(-228/24) = -9 -1/2. 

The sequence is shifted right by 9 positions in this case.