A261733 - OEIS

%I #12 Sep 01 2015 04:21:46

%S 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,

%T -9,10,-12,11,-13,15,-16,17,-18,22,-23,23,-27,30,-31,32,-35,40,-40,42,

%U -48,51,-54,57,-63,69,-71,78,-85,90,-97,102,-110,118,-124,133

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

%C 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.

%H Alois P. Heinz, <a href="/A261733/b261733.txt">Table of n, a(n) for n = 0..10000</a>

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

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*[0, -1, 0,

%p         -1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1]

%p        [1+irem(d, 18)], d=divisors(j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..80);  # _Alois P. Heinz_, Sep 01 2015

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

%Y 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).

%K sign

%O 0,9

%A _Vaclav Kotesovec_, Aug 30 2015