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A261733
Expansion of Product_{k>=1} (1 + x^(9*k))/(1 + x^k).
11
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
0,9
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
FORMULA
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)/3) / (2 * 3^(3/4) * n^(3/4)).
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]
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).
Sequence in context: A025829 A029285 A134337 * A268341 A053633 A216460
KEYWORD
sign
AUTHOR
Vaclav Kotesovec, Aug 30 2015
STATUS
approved