A261733 Expansion of Product_{k>=1} (1 + x^(9*k))/(1 + x^k).

1, -1, 0, -1, 1, -1, 1, -1, 2, -1, 1, -2, 2, -2, 2, -3, 4, -3, 4, -5, 5, -6, 6, -7, 8, -8, 9, -9, 10, -12, 11, -13, 15, -16, 17, -18, 22, -23, 23, -27, 30, -31, 32, -35, 40, -40, 42, -48, 51, -54, 57, -63, 69, -71, 78, -85, 90, -97, 102, -110, 118, -124, 133

Offset: 0

Author

Vaclav Kotesovec, Aug 30 2015

Keywords

• sign

Comments

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261735 (m=8), A145707 (m=10).

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*[0, -1, 0,
        -1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1]
       [1+irem(d, 18)], d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 01 2015

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^(9*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)/3) / (2 * 3^(3/4) * n^(3/4)).