A173030 Expansion of q^(-1/6) * (eta(q)^4 + 7 * eta(q^7)^4) in powers of q.

1, 3, 2, 8, -5, -4, -10, 8, -19, 0, 14, -16, -10, -4, 0, 6, 14, 20, 2, 0, -11, 20, 24, -16, 0, -4, 14, 8, -9, -15, 26, 0, 2, -28, 0, -16, -12, -28, -22, 0, 14, 16, 0, -30, 0, -28, 26, 32, -17, 0, 24, -16, -22, 0, -10, 32, -34, 55, 14, 0, 45, -4, 38, 8, 0, 0, -34, -8, 38, 0, -22, 42, 2, -28, 0, 0, -10, 20, -48, -40, -20, 44

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

Formula

Expansion of f(-x)^4 + 7 * x * f(-x^7)^4 = chi(-x) * chi(-x^7) * (psi(x)^4 + 7 * x^3 * psi(x^7)^4) in powers of x where psi(), chi(), f() are Ramanujan theta functions.

Expansion of (phi(-x)^4 + 7 * phi(-x^7)^4) / (8 * chi(-x) * chi(-x^7)) in powers of x^2 where phi(), chi(), f() are Ramanujan theta functions.

a(n) = b(6*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (-p)^(e/2) (1 + (-1)^e)/2 if p == 5 (mod 6).

G.f. is a period 1 Fourier series which satisfies f(-1 / (252 t)) = 252 (t/i)^2 * f(t) where q = exp(2 Pi i t).

A000727(n) = a(n) if n != 1 (mod 7). A000727(7*n + 1) + 7 * A000727(n) = a(7*n + 1).

Example

G.f. = 1 + 3*x + 2*x^2 + 8*x^3 - 5*x^4 - 4*x^5 - 10*x^6 + 8*x^7 - 19*x^8 + ...
G.f. = q + 3*q^7 + 2*q^13 + 8*q^19 - 5*q^25 - 4*q^31 - 10*q^37 + 8*q^43 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4 + 7 x QPochhammer[ x^7]^4, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 + 7 * x * eta(x^7 + A)^4, n))};

Crossrefs

Cf. A000727.

Sequence in context: A097018 A127541 A053219 * A271589 A083087 A191724

Adjacent sequences: A173027 A173028 A173029 * A173031 A173032 A173033

Keyword

sign

Author

Michael Somos, Feb 07 2010

Status

approved