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Expansion of Product_{k>0} (1 - x^(9*k)) * (1-x^(9*k-2)) * (1-x^(9*k-7)) / ((1-x^(9*k-1)) * (1-x^(9*k-6)) * (1-x^(9*k-8))).
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%I #10 Jan 09 2023 13:02:33

%S 1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0,0,0,0,1,0,0,1,1,0,0,0,0,1,0,0,1,

%T 0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,-1,0,1,1,0,1,1,0,0,

%U -1,0,1,0,0,1,1,0,-1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,1,0,0,0,0

%N Expansion of Product_{k>0} (1 - x^(9*k)) * (1-x^(9*k-2)) * (1-x^(9*k-7)) / ((1-x^(9*k-1)) * (1-x^(9*k-6)) * (1-x^(9*k-8))).

%C |a(n)|<2 if n<156, |a(n)|<3 if n<250.

%C It appears that a(n) = 0 if n == 2 (mod 3). - _Robert Israel_, Jan 31 2018

%H Robert Israel, <a href="/A143242/b143242.txt">Table of n, a(n) for n = 0..3000</a>

%F Euler transform of period 9 sequence [ 1, -1, 1, 0, 0, 0, -1, 1, -1, ...].

%F G.f.: Product_{k>0} (1 - x^(9*k)) * (1-x^(9*k-2)) * (1-x^(9*k-7)) / ((1-x^(9*k-1)) * (1-x^(9*k-6)) * (1-x^(9*k-8))).

%e 1 + q + q^3 + q^4 + q^6 + q^9 + q^12 + q^13 + q^16 + q^21 + q^24 + q^25 + ...

%p N:= 100: # to get a(0)..a(N)

%p g:= mul((1 - x^(9*k)) * (1-x^(9*k-2)) * (1-x^(9*k-7)) / ((1-x^(9*k-1)) * (1-x^(9*k-6)) * (1-x^(9*k-8))),k=1..floor((N+8)/9)):

%p S:= series(g,x,N+1):

%p seq(coeff(S,x,n),n=0..N); # _Robert Israel_, Jan 31 2018

%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^([1, -1, 1, -1, 0, 0, 0, 1, -1][k%9 + 1]), 1 + x * O(x^n)), n))}

%K sign,look

%O 0,157

%A _Michael Somos_, Aug 01 2008