A280666 Expansion of eta(q)^6/eta(q^6) in powers of q.

1, -6, 9, 10, -30, 0, 12, 36, 9, -60, -12, -54, 62, 120, 18, -72, -102, -54, -36, 156, 108, 48, -192, -108, 156, 78, 126, -206, -324, -72, 240, 324, 225, -168, -276, -180, 132, 264, 72, -144, -588, -198, 240, 804, 270, -288, -312, -324, 206, 486, 225, -528

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: Product_{n>0} (1-x^n)^6/(1-x^(6*n)).

Euler transform of period 6 sequence [ -6, -6, -6, -6, -6, -5, ...].

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/32) * Pi^(5/4) * 2^(5/6) * (sqrt(2) * (3^(1/2)-1))^(3/2) * (sqrt(2) * (1+3^(1/2)))^(5/3) / Gamma(3/4)^2 / Gamma(11/12)^(3/2) / Gamma(7/12)^(3/2) = A389010. - Simon Plouffe, Sep 22 2025

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d, 6)=0, -5, -6), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 07 2017

Mathematica

QP = QPochhammer; QP[x]^6/QP[x^6] + O[x]^70 // CoefficientList[#, x]& (* Jean-François Alcover, Mar 25 2017 *)

Prog

(PARI) q='q+O('q^66); Vec( eta(q)^6/eta(q^6) ) \\ Joerg Arndt, Mar 25 2017

Crossrefs

Cf. A002448 (k=2), A005928 (k=3), A083703 (k=4), A109064 (k=5), this sequence (k=6), A160534 (k=7).

Sequence in context: A029843 A209941 A000729 * A282937 A106248 A132725

Adjacent sequences: A280663 A280664 A280665 * A280667 A280668 A280669

Keyword

sign

Author

Seiichi Manyama, Jan 07 2017

Status

approved