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%I #12 Apr 28 2014 11:11:56
%S 1,-1,1,-1,0,1,-1,0,1,-2,2,0,-2,2,-1,-1,3,-2,-1,3,-3,0,4,-5,2,3,-6,4,
%T 2,-7,6,0,-7,9,-2,-7,10,-5,-6,13,-8,-5,15,-13,-1,16,-17,2,16,-22,8,16,
%U -27,14,12,-30,22,9,-34,29,3,-36,39,-5,-37,47,-14,-36,58,-26,-33,66,-41,-26,75,-56,-18,81,-74,-4,87,-94,12,87,-113,34
%N Let qf(a,q) = Product(1-a*q^j,j=0..infinity); g.f. is qf(q,q^3)/qf(q^2,q^3).
%H Alois P. Heinz, <a href="/A111165/b111165.txt">Table of n, a(n) for n = 0..10000</a>
%F Euler transform of period 3 sequence [ -1, 1, 0, ...]. - _Michael Somos_, Dec 23 2007
%F G.f.: Product_{k>=0} (1 - x^(3*k+1)) / (1 - x^(3*k+2)).
%p a:= proc(n) option remember; `if`(n=0, 1,
%p add(add(d*[0, -1, 1][irem(d, 3)+1],
%p d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Apr 02 2014
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*{0, -1, 1}[[Mod[d, 3]+1]], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Apr 28 2014, after _Alois P. Heinz_ *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=0, n\3, (1 - x^(3*k+1)) / (1 - x^(3*k+2)), 1 + x * O(x^n)), n))} /* _Michael Somos_, Dec 23 2007 */
%Y Cf. A111375. Convolution inverse of A111317.
%K sign,look
%O 0,10
%A _N. J. A. Sloane_, Nov 09 2005
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