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A083703 Expansion of eta(q)^4/eta(q^4) in powers of q.

%I A083703 #20 Apr 30 2017 04:49:45
%S A083703 1,-4,2,8,-4,-8,-8,16,6,-12,8,8,-8,-24,0,16,12,-16,10,24,-8,-16,-24,
%T A083703 16,8,-28,8,32,-16,-8,0,32,6,-32,16,16,-12,-40,-24,16,24,-16,16,40,-8,
%U A083703 -40,0,32,24,-36,10,16,-24,-24,-32,48,0,-32,24,24,-16,-40,0,48,12,-16,16,56,-16,-32,-48,16,30,-64,8,40,-24
%N A083703 Expansion of eta(q)^4/eta(q^4) in powers of q.
%C A083703 Euler transform of period 4 sequence [ -4,-4,-4,-3,...].
%H A083703 Seiichi Manyama, <a href="/A083703/b083703.txt">Table of n, a(n) for n = 0..10000</a>
%F A083703 G.f.: Product_{n>0} (1-x^n)^4/(1-x^(4n)).
%F A083703 a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A285895(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 29 2017
%p A083703 with(numtheory):
%p A083703 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A083703       `if`(irem(d, 4)=0, -3, -4), d=divisors(j))*a(n-j), j=1..n)/n)
%p A083703     end:
%p A083703 seq(a(n), n=0..80);  # _Alois P. Heinz_, Jan 07 2017
%t A083703 CoefficientList[QPochhammer[x]^4/QPochhammer[x^4] + O[x]^80, x] (* _Jean-François Alcover_, Sep 19 2016 *)
%o A083703 (PARI) a(n)=if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^4/eta(X^4),n))
%Y A083703 A080965(n)=(-1)^n a(n). a(2n)=0 iff n in A004215 (checked up to n=343).
%Y A083703 a(2n)=0 iff A005875(n)=0.
%Y A083703 Cf. A274327, A285895.
%K A083703 sign
%O A083703 0,2
%A A083703 _Michael Somos_, May 04 2003